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364  1911     , 
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The  teaching  of 


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University  of  California 

Los  Angeles 

Form  L-l 


This  book  is  DUE  on  the  last  date  stamped  below 


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JUL 


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Form  L-9-2w-7,'22 


OCT  6      1931 


FEB  1?  1932 


By  DAVID  EUGENE  SMITH,  LL.D. 

Professor  of  Mathematics  in  Teachers  College,  Columbia  University 


-2.  2.  &  d 


Reprinted,  with  revisions  and  additions,  from  the  Teachers  College 
Record,  Vol.  X,  No.  i,  January,  1909 


FOURTH  EDITION 


PUBLISHED  BY 

,  doliunbta 
NEW  YORK  CITY 
1911 


922 


Copyright,  1909,  by  Teachers  College,  Columbia  University 


PKESS  OF 

THE  BRANDOW  FEINTING  COMPANY 
ALBANT,  N.  Y. 


©A 


crp- 

PREFACE 

The  Teachers  College  Record  for  March,  1903,  contained  an 
article  on  Mathematics  in  the  Elementary  School  by  the  author's 
colleague  Professor  McMurry  and  himself.  This  number,  how- 
ever, has  long  since  been  out  of  print,  and  as  a  result  of  this  fact 
it  was  thought  best  in  the  autumn  of  1908  to  prepare  a  new 
number  of  the  Record  on  The  Teaching  of  Arithmetic.  This 
was  done  by  the  author,  and  the  article  appeared  in  January,  1909. 
Although  it  was  thought  that  the  unusually  large  edition  was 
sufficient  for  all  demands  for  some  years  to  come,  it  was  ex- 
hausted within  a  few  weeks,  and  it  became  necessary  to  print  the 
article  in  book  form.  In  spite  of  the  fact  that  the  work  was 
originally  written  in  a  popular  style,  to  the  end  that  it  might  be 
read  by  those  who  have  no  more  interest  in  mathematics  than 
in  the  various  other  subjects  of  the  curriculum,  it  has  been 
thought  best  to  make  but  a  few  changes  in  arranging  for  its 
publication  in  the  present  form.  Intended  as  it  is  for  those  who 
are  teaching  or  supervising  the  work  in  arithmetic  in  the  elemen- 
tary schools,  it  would  hardly  serve  its  purpose  if  it  departed 
widely  from  the  practical  and  entered  the  domain  of  pure  theory. 
As  between  influencing  the  few  or  the  many  on  a  topic  of  such 
general  interest  it  has  been  thought  better  to  adopt  the  latter 
course,  and  to  prepare  a  book  that  might  have  place  in  educational 
reading  circles  generally  and  serve  as  a  basis  for  the  work  in  the 
class-room  for  the  training  of  teachers. 


TABLE  OF  CONTENTS 


CHAPTER  I.  ^  The  History  of  the  Subject 5 

CHAPTER  II.  •  The  Reasons  for  Teaching  Arithmetic 8 

CHAPTER  III^  What  Arithmetic  should  include 12 

CHAPTER  IV.  The  Nature  of  the  Problems 15  • 

CHAPTER  V.  The  Arrangement  of  Material 19 

CHAPTER  VI.  Method 22 

CHAPTER  VII.  Mental  or  Oral  Arithmetic 26 

CHAPTER  VIII.  Written    Arithmetic    30 

CHAPTER  IX.  Children's  Analyses   35 

CHAPTER  X.  Interest  and  Effort 39 

CHAPTER  XI.  Improvements  in  the  Technique  of  Arithmetic 42 

CHAPTER  XII.  Certain  Great  Principles  of  Teaching  Arithmetic. .     52 

CHAPTER  XIII.  General  Subjects  for  Experiment 55 

CHAPTER  XIV.  Details  for  Experiment 65 

CHAPTER  XV.  The  Work  of  the  First  School  Year 76 

CHAPTER  XVI.  The  Work  of  the  Second  School  Year 85 

CHAPTER  XVII.  The  Work  of  the  Third  School  Year 91 

CHAPTER  XVIII.  The  Work  of  the  Fourth  School  Year 98 

CHAPTER  XIX.  The  Work  of  the  Fifth  School  Year 102 

CHAPTER  XX.  The  Work  of  the  Sixth  School  Year 107 

CHAPTER  XXL  The  Work  of  the  Seventh  School  Year m 

CHAPTER  XXII.  The  Work  of  the  Eighth  School  Year 115 


THE  TEACHING  OF  ARITHMETIC 


CHAPTER  I 

S/t>.  7-  1 

THE  HISTORY  OF  THE  SUBJECT 
2.  2.2.  8<D 

Of  all  the  sciences,  of  all  the  subjects  generally  taught  in  the 
common  schools,  arithmetic  is  by  far  the  oldest.  Long  before 
man  had  found  for  himself  an  alphabet,  long  before  he  first  made 
rude  ideographs  upon  wood  or  stone,  he  counted,  he  kept  his 
tallies  upon  notched  sticks,  and  he  computed  in  some  simple  way 
by  his  fingers  or  by  pebbles  on  the  ground.  He  did  not  always 
count  by  tens  as  in  our  decimal  system ;  indeed  this  was  a  rather 
late  device,  and  one  suggested  by  his  digits.  At  first  he  was 
quite  content  to  count  to  two,  and  generations  later  to  three,  and 
then  to  four.  Then  he  repeated  his  threes  and  had  what  we  call 
a  scale  of  three,  and  then,  as  time  went  on,  he  used  a  scale  of 
four,  and  then  a  scale  of  five.  At  one  time  he  seems  to  have 
used  the  scale  of  twelve,  because  he  found  that  twelve  is  divisible 
by  more  factors  than  ten,  and  particularly  by  two  and  three  and 
four;  but  by  the  time  he  became  ready  to  write  his  numbers 
the  convenience  of  finger  reckoning  had  become  so  generally 
recognized  that  ten  became  practically  the  universal  radix. 
Nevertheless  there  remain  in  our  language  and  customs  numerous 
relics  of  the  duodecimal  idea,  such  as  the  number  of  inches  in  a 
foot,  of  ounces  in  a  troy  pound,  and  of  pence  in  a  shilling,  all 
influenced  by  the  Roman  inclination  to  make  much  use  of  twelve 
in  practical  computation. 

The  writing  of  numbers  has  undergone  more  change  than 
even  the  number  names.  Not  only  was  there  a  notation  for  each 
language  in  ancient  times,  as  to-day  in  the  Orient,  but  some  lan- 
guages had  several  sets  of  numerals,  as  is  seen  in  the  three 
standard  systems  of  Egypt,  the  two  of  Greece,  and  the  somewhat 
varied  forms  in  use  in  Rome.  The  Roman  supremacy  gave  the 


6  The  Teaching  of  Arithmetic 

numerals  of  these  people  great  influence  in  Europe,  and  they  were 
practically  in  universal  use  in  the  West  until  the  close  of  the 
Middle  Ages.  Meantime  there  had  arisen  in  the  East,  probably 
in  India  although  very  likely  subjected  to  influence  from  without, 
our  present  system  of  notation,  and  little  by  little  this  permeated 
the  West.  When  it  arose  it  was  without  a  zero,  and  hence  with- 
out such  place  value  as  we  use  to-day;  but  probably  about  the 
seventh  century  the  zero  appeared,  and  the  completed  system 
found  its  way  northward  into  Persia  and  Arabia,  and  thence  in 
due  time  it  was  transmitted  to  the  West. 

At  first  the  subject  was  purely  practical,  a  counting  of  arrows 
or  of  sheep  or  of  men.  For  a  long  time  this  was  all  that  number 
meant  to  the  world,  until  the  mystic  age  developed  and  phil- 
osophy began.  Then  numbers  were  differentiated,  and  odd  and 
even  were  distinguished,  and  "  There's  luck  in  odd  numbers  " 
became  a  tenet  of  faith,  and  the  even  numbers  became  designated 
as  earthly  and  feminine.  The  story  is  long  and  interesting,  that 
of  the  development  of  this  mysticism  with  its  special  prominence 
of  three  and  seven,  particularly  so  as  the  movement  led  to  a 
study  of  the  properties  of  numbers,  to  roots,  and  to  series.  It 
is  connected  with  number  games,  and  these  in  turn  led  -to  the 
abacus,  and  so  the  practical  and  the  mysterious  are  more  or  less 
blended  even  at  times  when  they  are  generally  regarded  as  widely 
separated. 

The  growth  of  topics  of  arithmetic  is  also  an  interesting  sub- 
ject for  investigation.  We  say  that  there  are  four  fundamental 
operations,  although  once  there  was  only  one,  and  at  another 
time  the  world  recognized  as  many  as  nine.  We  operate  chiefly 
with  decimal  fractions,  as  in  working  with  dollars  and  cents, 
although  these  fractions  are  scarcely  three  hundred  years  old. 
We  are  impatient  that  a  child  stumbles  over  common  fractions, 
and  yet,  so  difficult  did  the  world  find  the  subject  that  for 
thousands  of  years  only  the  unit  fraction  was  used.  We  wonder 
how  the  long  division  form  of  greatest  common  divisor  ever  had 
place  in  arithmetic,  and  yet  it  was  a  practical  necessity  in  busi- 
ness until  about  1600  A.  D.  We  feel  that  "  partnership  involving 
time  "  could  never  have  been  practical,  and  yet  until  a  couple  of 
centuries  ago  it  was  intensely  so.  And  thus  it  is  with  many 
topics  of  arithmetic, — they  have  changed  from  century  to  cen- 


The  History  of  the  Subject  7 

tury,  and  even  in  our  own  time  from  year  to  year.  It  is  well  for 
a  teacher  to  know  a  little  of  this  history  of  the  subject  taught, 
although  space  does  not  allow  for  any  serious  consideration  of  the 
topic  in  this  work.  In  the  bibliography  some  reference  will  be 
found  to  sources  easily  available,  and  the  teacher  who  wishes 
to  see  arithmetic  in  progress,  as  opposed  to  arithmetic  stagnant 
and  filled  with  the  obsolete,  should  become  acquainted  with  one 
or  more  of  these  works  upon  the  subject.  The  history  of  arith- 
metic is  the  best  single  stimulus  to  good  method  in  teaching  the 
subject. 

BIBLIOGRAPHY:  Smith,  The  Teaching  of  Elementary  Mathe- 
matics, New  York,  1900;  Smith,  Rara  Arithmetica,  Boston, 
1909;  Smith  and  Karpinski,  Hindu-Arabic  Numerals,  Boston, 
1911;  Ball,  A  Primer  of  the  History  of  Mathematics,  Lon- 
don, 1895,  and  A  Short  Account  of  the  History  of  Mathematics, 
London,  4th  edition,  1908 ;  Fink,  History  of  Mathematics,  trans- 
lated by  Beman  and  Smith,  Chicago,  1900;  Cajori,  History 
of  Elementary  Mathematics,  New  York,  1896,  and  History 
of  Mathematics,  New  York,  1893 ;  Jackson,  The  Educational 
Significance  of  Sixteenth  Century  Arithmetic^  New  York, 
1906;  Gow,  A  Short  History  of  Greek  Mathematics,  Cam- 
bridge, 1884;  Conant,  The  Number  Concept,  New  York,  1896; 
Brooks,  Philosophy  of  Arithmetic,  revise!  edition,  Philadelphia, 
1902.  There  are  numerous  works  in  German  on  the  history  of 
mathematics  and  of  mathematical  teaching,  and  a  considerable 
number  in  French  and  Italian. 


CHAPTER  II 
THE  REASONS  FOR  TEACHING  ARITHMETIC 

The  ancients  had  less  difficulty  than  we  have  in-  assigning  a 
reason  for  teaching  arithmetic,  because  they  generally  differenti- 
ated clearly  between  two  phases  of  the  subject.  The  Greeks,  for 
example,  called  numerical  calculation  by  the  name  logistic,  and 
this  subject  was  taught  solely  for  practical  purposes  to  those  who 
were  going  into  trade.  A  man  might  have  been  a  very  good 
philosopher  or  statesman  or  warrior  without  ever  having  learned 
to  divide  one  long  number  by  another.  Such  a  piece  of  knowl- 
edge would  probably  have  been  looked  upon  as  a  bit  of  technical 
training,  like  our  use  of  the  slide  rule  or  the  arithmometer.  On 
the  other  hand  the  Greeks  called  their  science  of  numbers  arith- 
metic, a  subject  that  had  nothing  whatever  to  do  with  addition, 
subtraction,  multiplication,  or  division,  and  that  excluded  all 
applications  to  trade  and  industry.  This  subject  was  taught  to 
the  philosopher,  and  to  the  man  of  "  liberal  education  "  as  we  still 
call  him.  It  considered  questions  like  the  factorability  of  num- 
bers, powers  and  roots,  and  series, — topics  having  little  if  any 
practical  application  in  the  common  walks  of  life.  Therefore 
when  a  Greek  was  asked  why  he  taught  logistic,  his  answer  was 
definite :  It  is  to  make  a  business  man  able  to  compute  sufficiently 
for  his  trade.  If  he  was  asked  why  he  taught  arithmetic,  as  the 
term  was  then  used,  his  answer  was  still  fairly  a  unit :  I  teach  it 
because  it  makes  a  man's  mind  more  philosophic. 

In  the  present  day  we  have  a  somewhat  more  difficult  task 
when  we  attempt  to  answer  this  question.  Arithmetic  with  us  in- 
cludes the  ancient  logistic,  and  we  teach  the  subject  to  all  classes 
of  people: — to  one  who  will  become  a  day  laborer,  belonging 
to  a  class  that  never  in  the  history  of  the  world  studied  such  a 
subject  until  very  recently;  to  the  tradesman,  who  never  uses  or 
cares  to  use  the  chapter  on  prime  numbers ;  to  the  statesman, 
who  will  probably  have  little  opportunity  to  employ  logarithms 

3 


The  Reasons  for  Teaching  Arithmetic  9 

in  any  work  that  may  come  to  him;  to  the  clergyman,  to  whom 
the  metric  system  will  soon  be  merely  a  name ;  to  the  housewife, 
the  farmer,  and  to  all  those  who  travel  the  multifarious  walks  of 
our  complex  human  life.  For  us  to  tell  why  we  teach  the  Ameri- 
can arithmetic  to  all  these  people  is  by  no  means  so  easy  as  it  was 
for  the  Greek  to  answer  his  simple  question. 

In  general,  however,  we  may  say  that  as  we  have  combined 
the  ancient  logistic  (calculation)  and  arithmetic  (theory)  in  one 
subject,  so  we  have  combined  the  Greek  purposes,  and  that  we 
teach  this  branch  because  it  is  useful  in  a  business  way  to  every 
one,  and  also  because  it  gives  a  kind  of  training  that  other  sub- 
jects do  not  give.  "^ 

As  to  the  first  reason  there  can  be  no  question.  When  the 
great  mass  of  men  were  slaves  the  business  phase  was  not  so 
important ;  but  now  that  every  man  is  to  a  great  extent  his  own 
master,  receiving  money  and  spending  it,  some  knowledge  of 
calculation  is  necessary  for  every  American  citizen.  To  elaborate 
upon  this  point,  is  superfluous.  .There  is,  however,  one  principle 
that  should  guide  us  in  the  consideration  of  this  phase  of  the 
question :  Whatever  pretends  to  be  practical  in  arithmetic  should 
really  be  so.  We  have  no  right  to  inject  a  mass  of  problems  on 
antiquated  investments,  on  obsolete  forms  of  partnership,  on  for- 
gotten methods  of  mercantile  business,  or  on  measures  that  are 
no  longer  common,  and  make  the  claim  that  these  problems  are 
practical.  If  we  wish  them  for  some  other  purpose,  well  and 
good;  but  as  practical  problems  they  have  no  right  to  appear. 
To  set  up  a  false  custom  of  the  business  world  is  as  bad  as  to 
teach  any  other  untruth;  it  places  arithmetic  in  particular,  and 
education  in  general,  in  a  false  light  before  pupils  and  parents, 
and  is  unjustified  by  any  reason  that  we  can  adduce.  An  obso- 
lete business  problem  has  just  one  reason  for  being,  and  this 
reason  is  that  it  ha£  historical  interest.  We  can  secure  the  mental 
discipline  as  well  by  other  means,  and  we  have  no  right  to  handi- 
cap a  child's  mind  with  things  that  he  will  be  forced  to  forget 
the  minute  he  enters  practical  life. 

There  remains  the  side  of  mental  discipline,  which  I  have  else- 
where called,  for  want  of  another  term  and  following  various 
other  writers,  the  culture  side.  What  mental  training  does  a 
child  get  from  arithmetic  that  he  does  not  get  from  biology,  or 


io  The  Teaching  of  Arithmetic 

Latin,  or  music?  This  is  a  question  so  difficult  to  answer  that 
no  one  has  yet  satisfied  the  world  in  his  reply,  and  no  one  is 
liable  to  do  so.  There  have  been  elaborate  articles  written  to 
show  that  the  proper  study  of  arithmetic  has  an  ethical  value, 
though  exactly  what  there  is  in  the  subject  to  make  us  treat  our 
neighbor  better  it  is  a  little  difficult  to  say.  Others  have  said  that 
arithmetic,  through  its  very  rhythm,  has  an  aesthetic  value,  as  is 
doubtless  true ;  but  that  this  is  generally  realized,  or  that  it  serves 
to  make  us  more  appreciative  of  the  beautiful,  is  hardly  to  be 
argued  with  any  seriousness.  Still  others  have  felt  that  by  com- 
ing in  contact  with  exact  and  provable  truth  an  individual  sets 
for  himself  a  higher  standard  in  all  other  lines  of  work,  and  this 
again  is  probably  the  case,  although  the  measure  of  its  influence 
has  never  been  satisfactorily  accomplished.  And  to  these  reasons 
may  be  added  many  more,  such  as  the  training  of  a  deductive 
science,  although  elementary  arithmetic  is  to  a  large  extent  induc- 
tive; the  training  in  concentration,  although  the  untangling  of  a 
Latin  construction  requires  quite  as  close  attention  ;  the  exaltation 
of  mind  that  comes  from  the  study  of  numbers  that  may  increase 
or  decrease  indefinitely, — and  others  of  like  nature.  And  out  of 
it  all,  what  shall  we  say?  ,That  arithmetic  has  no  mental  disci- 
pline that  other  subjects  do  not  give?  No  one  really  feels  this, 
in  spite  of  the  fact  that  the  exact  nature  of  this  discipline  is  hard 
to  formulate.  Every  one  is  conscious  that  he  got  something  out 
of  the  study,  aside  from  calculation  and  business  applications, 
that  has  made  him  stronger,  and  the  few  really  scientific  investi- 
gations that  have  been  made,  as  to  the  effect  of  mathematical 
study,  bear  out  this  intuitive  feeling.  And  this  being  so,  we 
might  be  justified  even  if  we  did  not  attempt  to  define  just  what 
this  is,  any  more  than  we  should  attempt  too  seriously  to  define 
time,  or  love,  or  God,  or  eternity. 

Not  to  dismiss  the  mental  discipline  side  too  summarily,  how- 
ever, and  at  the  same  time  seeking  to  avoid  the  endless  verbiage 
that  usually  characterizes  the  discussion,  it  is  well  to  set  forth 
more  clearly  some  of  the  objects  to  be  sought  on  the  culture  side 
of  arithmetic.  In  the  first  place,  we  seek  an  absolute  accuracy 
of  operation  that  differs  from  the  kind  of  accuracy  we  seek  in 
science  or  linguistics  or  music.  The  fact  that  we  have,  in  thou- 
sands of  problems,  sought  a  result  so  exact  as  to  stand  every 


The  Reasons  for  Teaching  Arithmetic  n 

test,  leads  us  to  set  a  higher  standard  of  accuracy  in  all  lines 
than  we  could  have  set  without  it.  This  justifies  the  introduction 
of  any  part  of  theoretical  arithmetic  for  which  the  pupil  is  mentally 
ready.  It  is  one  reason  why  cube  root  was  formerly  there,  when 
pupils  were  more  mature  than  now,  and  in  the  same  way  it  has 
justified  progressions  and  a  more  elaborate  treatment  of  primes 
than  any  business  need  would  warrant.  Here  then,  is  a  reason 
for  teaching  arithmetic  that  is  above  and  beyond  the  merely 
practical  of  the  present  moment. 

A  similar  and  related  reason  appears  in  the  fact  that  mathe- 
matics in  general,  and  arithmetic  in  particular,  requires  a  helpful 
form  of  analysis  that  does  not  stand  out  so  clearly  in  other  studies. 
"  I  can  prove  this  if  I  can  prove  that ;  I  can  prove  that  if  I  can 
prove  a  third  thing ;  but  I  can  prove  that  third  thing ;  hence  I  see 
my  way  to  proving  the  first."  This  is  the  analytic  form  that  has 
come  down  to  us  from  Plato.  It  more  evidently  appears  in 
geometry,  but  is  essentially  the  reasoning  of  arithmetic  as  well. 
"  I  can  find  the  cost  of  2l/2  yd.  if  I  can  find  the  cost  of  I  yd. ;  but 
I  know  the  cost  of  6%  yd.,  so  I  can  find  the  cost  of  i  yd. ;  hence 
I  can  solve  my  problem,"  is  the  unworded  line  of  the  child's 
analysis.  Such  a  training,  unconsciously  received  and  often 
unconsciously  given,  is  valuable  in  every  problem  we  meet, 
leading  us  to  exclude  the  non-essential  -and  hold  with  tenacity 
to  a  definite  line  of  argument. 

These  two  phases  of  the  culture  side  of  arithmetic,  the  side 
of  mental  discipline,  will  then  suffice  for  our  present  purpose, 
which  is  to  show  that  such  a  side  exists:  (i)  The  contact  with 
absolute  truth;  (2)  The  acquisition  of  helpful  forms  of  analytic 
reasoning. 

BIBLIOGRAPHY:  Smith,  Teaching  of  Elementary  Mathematics, 
pp.  1-70;  Smith,  Teaching  of  Geometry,  Boston,  1911;  Suz- 
zallo,  Teaching  of^- Primary  Arithmetic,  Boston,  1911;  Young, 
The  Teaching  of  Mathematics,  New  York,  1907,  pp.  9,  41-52, 
202-256 ;  Bran  ford,  A  Study  of  Mathematical  Education,  Oxford, 
1908 ;  Stone,  Arithmetical  Abilities,  Teachers  College  Series, 
1908;  Rietz  and  Shade,  Correlation  of  Efficiency  in  Mathe- 
matics, etc.,  University  of  Illinois  Bulletin,  vol.  VI,  no.  10;  Hill, 
Educational  Value  of  Mathematics,  Educational  Review,  vol.  IX, 
p.  349 ;  Schubert,  Mathematical  Essays  and  Recreations,  Chicago, 
1899,  p.  27. 


CHAPTER  III 
WHAT  ARITHMETIC  SHOULD  INCLUDE 

If  we  taught  arithmetic  only  for  its  utilitarian  value,  to  fit  a 
person  for  the  computations  that  the  average  man  needs  to  per- 
form or  know  about  in  daily  life,  the  range  of  subject  matter 
would  not  be  great.  Addition,  particularly  of  money,  but  not 
involving  very  large  or  numerous  amounts,  is  probably  the  most 
important  topic.  Perhaps,  for  it  is  difficult  to  say  with  certainty, 
the  simple  fractions  ^  and  *4  afe  next  in  line  of  relative  impor- 
tance, including  y2  of  a  sum  of  money,  l/4  of  a  length  or  weight, 
and  so  on.  Very  likely  the  making  of  change,  one  of  the  forms 
of  subtraction,  is  next  in  frequency  of  use.  Then  may  come  easy 
multiplications,  to  find  the  cost  of  5  Ib.  of  sugar,  or  of  16  yd.  of 
cloth,  given  the  price  per  pound  or  yard.  A  few  of  the  most 
commonly  used  measures  and  their  relations  must  then  be  known, 
as  that  12  inches  equal  i  foot  and  16  ounces  equal  i  pound. 
Given  this  equipment,  the  average  run  of  humanity  would  be  able 
to  get  along  fairly  well.  But  beyond  this  there  lies  a  second  field 
of  work  that  every  one  may  need,  that  a  large  minority  will  need, 
and  that  we  must  all  at  least  know  something  about.  This  field 
includes  all  four  fundamental  operations  with  integers,  with  simple 
common  fractions  (say  with  denominators  of  one  or  two  figures), 
with  decimal  fractions  at  least  to  hundredths,  and  with  compound 
numbers  of  at  least  two  denominations ;  the  common  business 
cases  of  percentage,  and  their  applications ;  the  common  problems 
of  business,  all  of  which  are  applications  of  the  operations  above 
mentioned,  and  a  little  knowledge  of  ratio  and  proportion,  chiefly 
for  understanding  the  meaning  of  these  terms.  From  the  stand- 
point of  business  needs  this  equipment  would  answer  the  purposes 
of  nearly  every  one.  Whatever  of  applied  arithmetic  lies  beyond 
this  is  a  part  of  the  technical  training  of  a  very  small  minority. 
Apothecary's  measures  form  part  of  the  technical  training  of  the 
drug  clerk  and  the  physician ;  the  average  citizen  has  long  since 

12 


What  Arithmetic  Should  Include  13 

forgotten  them,  and  happily  so.  Compound  proportion  is  never 
used  practically,  and  any  mathematician  if  called  upon  to  solve 
its  problems  would  employ  another  and  a  better  method.  Duo- 
decimals, while  interesting  historically  and  philosophically,  from 
the  practical  standpoint  are  used  by  so  few  as  to  place  them 
also  in  the  technical  training  of  the  very  small  minority.  Subjects 
like  discount  and  interest  are,  of  course,  included  under  the  com- 
mon applications  of  percentage.  Similarly  with  stocks  and  bonds, 
for  although  such  securities  are  purchased  by  relatively  few 
people,  their  nature  and  uses  should  be  understood  by  all,  par- 
ticularly as  we  seem  to  have  entered  upon  an  era  of  extensive 
cooperation  upon  a  stock  basis.  The  general  nature  of  applica- 
tions, however,  will  be  discussed  later. 

If  we  taught  arithmetic  only  from  the  standpoint  of  mental 
discipline  we  might  use  all  the  material  here  mentioned,  and  any 
other  topics  that  allowed  for  securing  accurate  results  by  clear 
reasoning  processes.  Obsolete  measures,  obsolete  methods,  pro- 
gressions, cube  root  and  even  higher  roots,  compound  proportion, 
— all  such  topics  might  have  place  if  we  were  seeking  only  the 
discipline  of  arithmetic.  When,  however,  we  consider  that  we 
are  seeking  to  unite  these  two  considerations,  and  are  attempting 
to  make  the  subject  both  practical  and  disciplinary,  then  we  are 
met  by  the  necessity  for  mutual  concessions.  The  practical  side 
must  concede  to  the  disciplinary  by  having  its  processes  clearly 
understood^  and  by  developing  the  reason  at  every  step ;  the 
disciplinary  side  must  concede  to  the  practical  by  selecting  its 
topics  in  such  way  as  to  give  no  false  notions  of  business,  and 
as  to  encourage  the  pupils  to  an  interest  in  the  quantitative  side 
of  the  world  about  them.  On  the  one  side  we  must  not  teach 
business  arithmetic  by  mere  arbitrary  rules  that  are  not  under- 
stood, since  this  would  be  to  eliminate  the  disciplinary  nature; 
on  the  other  side  we  must  not  introduce  a  style  of  time  draft  that 
is  now  obsolete  in  America,  or  artificial  examples  in  compound 
proportion,  because  these  inculcate  wrong  ideas  of  the  business 
world  about  us,  nor  extensive  work  in  equation  of  payments 
because  this  is  part  of  the  technical  training  of  such  a  very 
small  minority  that  we  can  use  our  time  to  better  advantage  by 
dwelling  upon  other  topics. 

A  word  should  also  be  said  as  to  the  tendency  of  some  teachers 


14  The  Teaching  of  Arithmetic 

to  feel  that  worthy  results  are  attained  when  children  have  been 
drilled  to  unnecessary  facility  in  one  line  or  another.  It  is  not  diffi- 
cult to  train  children  to  add  two  columns  of  figures  at  a  time, 
or  to  multiply  by  cross  multiplication,  or  to  detect  six  or  eight 
cubes  at  a  glance  or  to  display  various  other  forms  of  arithmetical 
ability  analogous  to  the  acrobatical  feats  sometimes  allowed  in 
a  gymnasium.  Such  activities  have  some  value  as  games,  and 
they  attract  attention  on  the  part  of  visitors,  but  it  is  a  question 
whether  the  pupil  derives  any  real  benefit  from  most  of  this  kind 
of  work.  It  is  because  of  this  feeling  that  such  features  are  not 
found  in  our  textbooks,  the  space  being  devoted  to  those  things 
that  the  business  man  finds  useful  in  everyday  life. 

Thus  it  happens  that  the  modern  American  arithmetic  is  a 
fair  compromise  between  the  practical  without  theory  and  the 
theoretical  without  practice,  the  two  distinct  phases  of  the  old 
Greek  number  work.  To  keep  this  balance  true  is  one  of  the 
missions  of  teachers  to-day.  The  tendency  is  to  obtain  the  mental 
discipline  of  arithmetic  from  problems  that  are  practical,  and  that 
this  tendency  is  a  healthy  one  there  seems  to  be  no  room  for 
reasonable  doubt. 

BIBLIOGRAPHY:  Smith,  Teaching  of  Elementary  Mathematics, 
p.  19 ;  Young,  pp.  23-242 ;  Spencer,  The  Teaching  of  Elementary 
Mathematics  in  the  English  Public  Elementary  Schools,  London, 
1911. 


CHAPTER  IV 
THE  NATURE  OF  THE  PROBLEMS 

In  no  way  has  arithmetic  changed  as  much  of  late  years  as  in 
the  nature  of  the  problems  and  the  arrangement  of  the  material. 
The  former  has  come  about  from  two  causes,  (i)  the  needs  of 
society,  and  (2)  the  study  of  child  psychology.  The  latter,  the 
arrangement  of  the  material,  has  been  determined  almost  entirely 
by  psychological  considerations.  In  this  chapter  it  is  proposed  to 
speak  of  the  former,  the  nature  of  the  problems. 

It  should  definitely  be  stated,  however,  that  this  emphasis  laid 
upon  applied  problems  should  not  be  construed  to  mean  that  the 
abstract  number  work  is  not  quite  so  important  as  the  concrete. 
To  be  sure  we  have  some  advocates  of  only  the  concrete,  what  the 
Germans  call  the  "  clothed  problem,"  to  the  entire  exclusion  of 
the  abstract  drill  work  of  the  Pestalozzi  school.  Such  extremists 
are,  however,  not  numerous,  and  they  have  but  a  small  following. 
Every  one  who  looks  into  the  subject  is  aware  that  on  the  score 
of  interest  a  child  prefers  to  work  with  abstract  numbers,  while 
as  to  the  final  results  upon  his  education  we  seem  to  neglect 
altogether  too  much  the  ability  to  get  exact  results  quickly  and 
with  a  certainty  as  to  their  exactness.  This  phase  of  the  work 
will  be  mentioned  in  a  later  chapter,  and  for  the  time  being  we 
may  well  consider  the  applied  problem. 

Within  the  last  few  years  the  question  of  the  practical  uses  of 
arithmetic  has  been  a  vital  one  in  educational  circles,  especially 
in  Germany  and  America,  resulting  in  considerable  literature 
upon  the  subject.  These  needs,  while  generally  similar  in  various 
countries,  differ  more  or  less  in  details.  Thus  a  country  whose 
business  was  chiefly  farming  would  need  to  emphasize  agricultural 
problems ;  one  that  derived  its  wealth  from  its  metals  or  its  coal 
would  emphasize  mining;  a  manufacturing  nation  would  find 
certain  lines  of  problems  of  the  factory  peculiarly  suited  to  its 
needs,  while  one  that  derived  its  wealth  chiefly  from  shipping 

15 


16  The  Teaching  of  Arithmetic 

would  require  those  relating  to  foreign  commerce.  The  mathe- 
matical foundation  would  be  the  same  in  all  cases,  but  the  material 
content  of  the  problem  would  vary.  Now  in  America  we  are 
unusually  cosmopolitan  in  our  needs  in  this  respect,  ranking  high 
in  all  these  particulars  save  only  (at  present)  in  ocean  traffic. 
We  are  therefore  very  fortunate  in  having  at  our  disposal  unlim- 
ited problem  material  that  relates  to  our  wide  range  of  national 
resources  and  industries.  The  advantage  of  using  this  material 
instead  of  the  obsolete  inherited  problems  that  came  down  to  us 
from  Italy,  through  England,  ought  to  be  so  evident  as  to  require 
no  argument.  There  will  always  be  some  who  cry  out  against 
what  they  call  encyclopedic  information  in  an  arithmetic,  but 
surely  if  a  problem  is  to  contain  any  facts  at  all  it  is  better  that 
these  facts  be  American  and  of  the  twentieth  century  than  Italian 
and  of  the  fifteenth.  Can  there  be  any  doubt  that  an  American 
boy  or  girl  will  get  more  breadth  of  view,  more  interest,  and 
possibly  more  directly  useful  information  from  a  problem  about 
the  mixing  of  plant  foods  for  a  southern  farm,  than  one  about 
the  mixing  of  teas  that  are  never  mixed  in  the  way  the  text-book 
says?  If  a  pupil  is  to  study  about  goods  being  transported,  is 
it  not  better  for  him  to  take  a  practical  case  relating  to  our  rail- 
roads than  the  old-time  one  of  pedlars  carrying  their  packs?  It 
is  well,  however,  to  avoid  the  unfortunate  tendency  manifested 
by  some  recent  writers  to  introduce  problem  material  that  no 
child  is  ready  to  understand  and  with  which  teachers  themselves 
should  not  be  expected  to  be  familiar.  Technical  information 
of  trades,  scientific  nomenclature  that  belongs  to  the  college  or  to 
the  later  years  in  the  high  school,  problems  of  the  civil  engineer 
or  the  food  chemist, — these  have  no  more  place  in  the  arithmetic 
class  than  has  the  apothecary's  table  or  the  subject  of  equation 
of  payments.  Common  information,  the  subjects  of  interest  to 
the  general  public,  and  those  matters  that  are  topics  of  conversa- 
tion in  the  usual  walks  of  life  are  the  bases  upon  which  we  may 
reasonably  build  our  problems.  Undertaken  in  this  spirit  we  need 
not  fear  if  critics  accuse  us  of  making  our  schools  encyclopedic. 
Every  usable  school  arithmetic  has  always  been  an  encyclopedia ; 
what  we  have  to  determine  is  whether  it  shall  now  be  an  encyclo- 
pedia of  vital,  modern  facts,  or  one  of  obsolete,  dull,  useless 
information.  The  needs  of  society  demand  the  former ;  vis  inertia 


17 

holds  to  the  latter.  The  earnest  teacher,  awake  to  the  needs  of  the 
business  community  in  which  a  school  is  located,  can  hardly  fail 
to  introduce  genuine  problems  with  local  color  to  enliven  the 
work  in  arithmetic.  No  text-book  can  fit  the  needs  of  every 
locality,  and  original  problems  are  easily  found  by  the  children 
themselves  if  the  opportunity  is  given.  The  awakening  of  inter- 
est in  such  work,  however,  will  not  come  from  the  style  of  text- 
book of  a  generation  ago;  it  will  only  come  in  connection  with 
the  study  of  a  book  that  is  itself  filled  with  this  spirit.  Fortu- 
nately most  of  our  modern  writers  are  working  earnestly  to 
meet  the  needs  of  to-day,  and  our  American  arithmetics  may 
well  lay  claim  to  being  among  the  most  progressive  that  are 
appearing  in  this  generation. 

But  what  as  to  the  effect  of  the  study  of  child  psychology? 
Here  too  there  has  been  made  very  great  progress  in  recent  years. 
Although  a  problem  may  represent  all  that  business  needs  suggest, 
it  still  may  not  be  suited  to  a  particular  school  year.  In  other 
words,  we  have  to  consider  from  grade  to  grade  the  interests  and 
powers  of  the  child.  We  would  not  think  of  giving  to  a  child 
in  the  first  grade  the  problem,  If  A  has  2  shares  of  railroad  stock 
and  B  has  3  shares,  how  many  have  they  together?  For  while 
the  child  can  add  2  and  3,  he  has  no  knowledge  of  stocks,  nor 
any  interest  in  them.  Change  the  subject  to  marbles  or  apples 
or  tops,  and  it  is  suited  to  his  mind,  but  not  otherwise.  Thus 
it  has  come  about  that  teachers  are  trying  to  decide  what  are  the 
larger  interests,  actual  or  potential,  of  the  children  in  the  various 
school  years,  to  the  end  that  the  problems  of  arithmetic  may  be 
the  better  apperceived.  A  beginning  has  been  made,  but  the 
future  will  see  the  work  extended.  We  know  that  pride  in  our 
national  resources  renders  interesting  a  style  of  problem  in  the 
fifth  school  year  that  would  be  of  no  value  in  the  second,  even 
with  smaller  numbers.  On  the  other  hand  we  are  equally  aware 
that  certain  problems  involving  children's  games  that  are  part  of 
the  genuine  applied  mathematics  of  the  third  year  would  have  no 
interest  whatever  in  the  eighth.  And  so  in  general,  teachers  are 
seriously  attempting  at  the  present  time  to  coordinate  the  inter- 
ests of  children,  the  needs  of  society,  and  the  mathematical  pow- 
ers in  each  of  the  grades  from  the  first  year  through  the  ele- 
mentary school. 


i8  The  Teaching  of  Arithmetic 

BIBLIOGRAPHY:  Smith,  Teaching  of  Elementary  Mathematics, 
p.  21 ;  Smith,  Teaching  of  Geometry,  Boston,  1911;  Young, 
pp.  97,  210.  Consult  also  the  author's  text-books  on  arithmetic 
and  works  like  Saxelby,  Practical  Mathematics,  London,  1905 ; 
Consterdine  and  Andrew,  Practical  Arithmetic,  London,  1907; 
Consterdine  and  Barnes,  Practical  Mathematics,  London,  1908; 
Castle,  Workshop  Mathematics,  London,  1907;  Castle,  Manual 
of  Practical  Mathematics,  London,  1906;  Cracknell,  Practical- 
Mathematics,  4th  edition,  London,  1906.  For  statistics  for  prob- 
lems The  World  Almanac,  New  York,  is  an  inexpensive  and 
valuable  source. 


CHAPTER  V 
THE  ARRANGEMENT  OF  MATERIAL 

As  already  stated,  the  two  most  noteworthy  changes  in  arith- 
metic in  recent  years  have  related  to  the  nature  of  the  problems 
and  the  arrangement  of  material.  The  latter  has  been  the  result 
of  a  more  or  less  serious  study  of  child  psychology,  namely  of 
the  powers  of  the  individual  in  the  various  school  years. 

Formerly,  say  a  century  or  more  ago,  it  was  the  custom  to 
study  arithmetic  from  a  single  book,  after  the  boy  (for  the  girl 
seldom  understood  mathematics  of  any  kind)  could  read  and 
write.  The  child  was  mature  enough  to  understand  the  subject 
after  a  fashion,  and  he  "  went  through  "  the  book.  But  as  arith- 
metic began  to  work  its  way  down  to  the  earliest  grades  it  was 
found  impracticable  to  follow  this  plan,  the  subject  being  too 
difficult  for  young  minds.  The  ordinary  text-book  was  therefore 
preceded  by  a  primer  on  arithmetic,  and  thus  a  two-book  series 
was  formed.  From  this  beginning  numerous  experiments  have 
proceeded,  seeking  to  carry  the  improvement  still  farther.  We 
have  had  three-book  series,  eight-book  series,  lesson  leaflets,  two- 
book  courses  arranged  by  grades,  and  so  on.  We  have  had 
spirals  of  various  degrees  of  turning,  efforts  to  resume  the 
topical  arithmetic,  books  arranged  to  follow  certain  narrow  lines 
of  manual  training,  and  so  on, — all  serious  efforts  for  better- 
ment, but  many  of  them  too  hastily  considered  to  have  any 
material  influence. 

In  the  midst  of  it  all,  what  have  the  practical  teachers  of  the 
country  done?  In  every  city,  in  several  states,  and  by  numerous 
associations,  courses  of  study  have  been  arranged  setting  forth 
the  material  that  experience  has  shown  can  be  used  to  advantage 
in  the  several  school  years,  and  selections  have  been  made  from 
the  current  arithmetics  to  supply  what  was  needed.  In  other 
words  the  practical  teachers  of  America  arranged  their  own 
books  to  a  great  extent,  basing  their  selection  of  material  upon 

19 


2O  The  Teaching  of  Arithmetic 

an  empirical  psychology,  and  giving  to  the  child  what  his  inter- 
ests and  capabilities  suggested. 

It  is  out  of  such  a  movement,  spontaneous  but  thoroughly 
sound,  that  the  later  American  text-books  have  arisen,  and  the 
care  and  earnestness  given  to  their  preparation  by  various  authors 
and  publishers  should  have  the  commendation  of  all. 

These  books  are  and  probably  will  continue  to  be  of  two  dis- 
tinct types,  each  with  strong  merits  of  its  own,  and  each  capable 
of  producing  the  best  work.  First  there  is  the  topical  arithmetic, 
that  is,  a  book  arranged  by  topics,  a  subject  like  percentage  being 
studied  once  for  all,  the  pupil  staying  with  it  until  it  is  thoroughly 
mastered.  Such  a  book  has  two  great  merits;  it  tends  to  keep 
the  child  upon  each  subject  long  enough  to  give  him  a  feeling 
of  mastery  that  he  would  not  have  if  he  studied  some  of  the 
scrappy  books  constructed  on  the  extreme  spiral  system,  and  it 
allows  a  teacher  who  wishes  to  adopt  a  moderate  spiral  to  do  this 
in  a  manner  that  will  meet  the  local  conditions.  By  this  latter 
is  meant  that  some  classes  seem  to  need  more  work  in  a  subject 
like  simple  interest  than  other  classes  do,  when  it  is  first  taken 
up.  A  teacher  may,  therefore,  select  as  much  or  as  little  as  is 
necessary  when  a  topic  is  taken  up  for  the  first  or  the  second 
time,  and  this  is  perhaps  more  easily  done  from  a  topical  than 
from  another  form  of  book. 

The  second  general  type  of  text-book  is  one  that  attempts  to 
fit  the  course  of  study  in  a  large  number  of  schools.  In  general 
these  books  agree  in  a  number  of  particulars:  (i)  Certain  sub- 
jects, such  as  the  most  commonly  used  operations,  should  appear 
several  times  in  the  school  course;  (2)  others,  such  as  the  busi- 
ness application  of  simple  interest,  may  appear  perhaps  two  or 
three  times,  with  gradually  increasing  difficulty;  (3)  still  others, 
like  board  measure,  may  be  sufficiently  treated  once  for  all ;  (4) 
the  closing  year  of  arithmetic  should  be  devoted  to  a  study  of 
such  higher  problems  of  business  as  can  be  understood  by  the 
children.  These  are  some  of  the  articles  of  agreement,  and  they 
go  to  show  the  common-sense  principles  on  which  our  modern 
courses  of  study  and  text-books  in  arithmetic  are  based. 

As  to  which  of  these  two  types  is  the  better  it  is  impossible 
to  give  a  general  decision.  It  depends  largely  upon  the  school 
and  the  teacher.  With  the  one  book  the  teacher  arranges  the 


The  Arrangement  of  Material  21 

matter  to  suit  the  pupils'  needs  ;  in  the  other  the  matter  is  already 
arranged  to  suit  the  average  pupil.  Neither  will  fit  each  indi- 
vidual case,  and  the  great  thing  is  not  so  much  the  arrangement 
of  the  matter  as  it  is  the  modern  spirit  displayed  in  the  omission 
of  the  obsolete  and  the  introduction  of  new,  vital,  interesting, 
intelligible  problems  of  to-day. 

It  should  also  be  stated  in  this  connection  that  a  similar  move- 
ment is  taking  place  with  reference  to  algebra  and  geometry,  and 
that  it  is  destined  to  reach  still  further  up  the  grade  and  into 
college  mathematics.  Algebra  is  old,  but  algebra  text-books  are 
relatively  modern.  They  were  at  first  based  upon  the  arithmetics 
that  preceded  them,  and  as  far  as  possible  they  followed  their 
arrangement  of  matter.  The  ordinary  school  algebra  is,  there- 
fore, merely  an  old-style  arithmetic  with  letters  used  instead  of 
numerals,  and  with  a  considerable  number  of  its  problems  taken 
from  the  arithmetical  collections  of  earlier  days.  The  arrange- 
ment of  matter  is  confessedly  not  scientific,  and  we  are  even  now 
seeing  the  sequence  of  topics  challenged,  and  a  serious  effort  to 
improve  the  nature  of  the  problems.  The  next  move  of  impor- 
tance in  the  teaching  of  elementary  mathematics  will  be  the  re- 
arrangement of  the  material  and  the  enriching  of  the  applications 
of  algebra  and,  probably  to  a  less  degree,  of  geometry. 

BIBLIOGRAPHY:  Consult  the  latest  text-books,  comparing  the 
two  types  and  noting  the  distinctive  advantages  of  each.  Consult 
also  those  extreme  forms  in  which  the  spiral  arrangement  is 
carried  too  far.  This,  and  in  general  the  other  questions  con- 
sidered in  this  monograph,  will  be  found  discussed  in  Professor 
Suzzallo's  noteworthy  work,  Teaching  of  Primary  Arithmetic, 
Boston,  1911. 


CHAPTER  VI 
METHOD 

Of  all  the  terms  used  in  educational  circles  "  Method  "  is  per- 
haps the  most  loosely  defined.  Efforts  have  been  made  to  limit 
its  meaning,  to  divide  its  responsibilities  with  such  terms  as 
"  Mode  "  and  "  Manner,"  but  it  still  stands  and  is  likely  to  stand 
as-  a  convenient  name  for  all  sorts  of  ideas  and  theories  and 
devices.  Nevertheless  it  has  been  most  often  used  in  arithmetic 
to  speak  of  the  general  plan  of  some  individual  for  introducing 
the  subject,  as  when  we  speak  of  the  Pestalozzi  or  Tillich  or 
Grube  Methods,  although  it  is  also  applied  to  such  arrangements 
of  material  as  is  indicated  by  the  expressions  Topical  Method 
and  Spiral  Method,  and  to  such  an  emphasis  of  some  particular 
feature  as  has  given  name  to  the  Ratio  Method.  It  is  not  the 
intention  to  attempt  any  definition  of  the  term  that  shall  include 
all  of  these  ramifications,  but  to  take  it  as  it  stands,  to  characterize 
briefly  some  of  these  "  Methods,"  and  then  to  speak  briefly  of  the 
subject  as  a  whole. 

Pestalozzi's  method  was  really  a  creation  of  his  followers. 
What  this  great  master  did  for  arithmetic  was  to  introduce  it 
much  earlier  into  the  school  course,  to  use  objects  more  system- 
atically to  make  the  number  relations  clear,  to  abandon  arbitrary 
rules,  to  drill  incessantly  on  abstract  oral  work,  and  to  emphasize 
the  unit  by  considering  a  number  like  6  as  "  6  times  I."  For  the 
time  in  which  he  lived  (about  1800)  all  this  was  a  healthy  protest 
against  the  stagnant  education  that  he  found.  To-day  it  is  only 
an  incidental  lesson  to  the  teacher,  although  Pestalozzi's  spirit 
and  several  of  his  ideas  may  well  command  the  admiration  and 
respect  of  all  who  study  the  results  of  his  great  work. 

The  "  Method  "  of  Tillich,  who  followed  Pestalozzi  by  a  few 
years  consisted  largely  of  making  a  systematic  use  of  sticks  cut 
in  various  lengths,  say  from  i  inch  to  10  inches.  It  is  evident 
that  such  a  collection  allowed  for  emphasizing  the  notion  of  tens, 

22 


Method  23 

for  treating  fractions  as  ratios,  and  for  visualizing  in  a  very  good 
way  the  simpler  number  relations.  On  the  other  hand  it  is  also 
evident  that  the  use  of  only  a  single  kind  of  material  is  based 
upon  a  much  narrower  idea  than  that  of  Pestalozzi,  who  pur- 
posely made  use  of  as  wide  a  range  of  material  as  he  could. 

The  Grube  Method,  that  created  such  a  stir  in  America  a 
generation  ago,  was  not  very  original  with  Grube  (1842).  Es- 
sentially it  was  an  adaptation  of  the  concentric  circle  plan  that 
had  already  been  used,  a  kind  of  spiral  arrangement  of  matter  to 
meet  the  growing  powers  of  the  child.  It  contained  some  absurd- 
ities, such  as  the  exhausting  of  the  work  on  one  number  before 
proceeding  to  the  next,  as  of  studying  25  in  all  its  relations  before 
learning  26 ;  but  on  the  whole  it  made  somewhat  for  progress  by 
assisting  to  develop  a  sane  form  of  the  spiral  idea. 

It  is  not  worth  while  to  speak  of  other  individual  "  Methods," 
since  they  have  little  of  value  to  the  practical  teacher,  and  the 
student  of  the  history  of  education  can  easily  have  access  to  them. 
Enough  has  been  said  to  show  that  one  of  the  easiest  things  in 
the  teaching  of  arithmetic  is  the  creation  of  "  Method  " — and  one 
of  the  most  useless.  We  may  start  off  upon  the  idea  that  all 
number  is  measure,  and  hence  that  arithmetic  must  consist  of 
measuring  everything  in  sight, — and  we  have  a  "  Measuring 
Method."  It  will  be  a  narrow  idea,  we  shall  neglect  much  that  is 
important,  but  if  we  put  energy  back  of  it  we  shall  attract  attention 
and  will  very  likely  turn  out  better  computers  than  a  poor  teacher 
will  who  is  wise  enough  to  have  no  "  Method,"  in  this  narrow 
sense  of  the  term.  Again,  we  may  say  that  every  number  is  a 
fraction,  the  numerator  being  an  integral  multiple  of  the  denomi- 
nator in  the  case  of  whole  numbers.  From  this  assumption  we 
may  proceed  to  teach  arithmetic  only  as  the  science  of  fractions. 
It  will  be  hard  work,  but,  given  enough  energy  and  patience  and 
skill,  the  children  will  survive  it  and  will  learn  more  of  arithmetic 
than  may  be  the  case  with  listless  teaching  on  a  better  plan.  We 
might  also  start  with  the  idea  that  every  lesson  should  be  a  unit, 
and  that  in  it  should  come  every  process  of  arithmetic,  so  far  as 
this  is  possible,  and  we  could  stir  up  a  good  deal  of  interest  in  our 
"  Unit  Method."  Or,  again,  we  could  begin  with  the  idea  that 
all  action  demands  reaction,  and  that  every  lesson  containing 
addition  should  also  contain  subtraction ;  that  6  +  4  =  10  should 


24  The  Teaching  of  Arithmetic 

be  followed  by  10  —  6  =  4  and  10  —  4  —  6 ;  and  that  2  X  5  =  10 
should  be  followed  by  10  -r-  2  —  5  and  10  -r-  5  =  2.  By  sufficient 
ingenuity  a  very  taking  scheme  could  be  evolved,  and  the  "  Inverse 
Method  "  would  begin  to  make  a  brief  stir  in  the  world.  This  in 
fact  has  been  the  genesis,  rise,  and  decline  of  Methods ;  given  a 
strong  but  narrow-minded  personality,  with  some  little  idea  such 
as  those  above  mentioned ;  this  idea  is  exploited  as  a  panacea ;  it 
creates  some  little  stir  in  circles  more  or  less  local ;  it  is  tried  in  a 
greater  or  less  number  of  schools ;  the  author  and  his  pupils  die, 
and  in  due  time  the  Method  is  remembered,  if  at  all,  only  by  some 
inscription  in  those  pedagogical  graveyards  known  as  histories  of 
education. 

The  object  in  writing  thus  is  manifest.  For  the  teacher  with 
but  little  experience  there  is  a  valuable  lesson,  namely,  that  there 
is  no  "  Method  "  that  will  lead  to  easy  victory  in  the  teaching  of 
arithmetic.  There  are  a  few  great  principles  that  may  well  be 
taken  to  heart,  but  any  single  narrow  plan  and  any  single  line  of 
material  must  be  looked  upon .  with  suspicion.  Certain  of  the 
general  principles  of  Pestalozzi  are  eternal,  but  the  Reckoning- 
chest  of  Tillich  is  practically  forgotten. 

And  here  it  is  proper  to  say  a  word  as  to  what  schools  of 
observation  and  practice  should  stand  for  in  these  matters  of 
method  and  purpose.  It  would  be  a  very  easy  thing  to  concentrate 
on  some  single  point,  some  device  of  teaching,  some  particular  line 
of  problems,  and  to  carry  the  work  to  an  extreme  that  would 
attract  attention  and  produce  results  that  would  be  remarked 
upon.  This  is  the  temptation  of  those  who  direct  such  schools. 
But  is  it  a  wise  policy?  These  schools  are  established  to  train 
teachers  to  well-balanced  leadership,  not  to  be  extremists  when 
this  means  the  neglect  of  essential  features  of  education.  The 
graduates  should  know  the  best  that  there  is  in  every  theory  of 
education,  but  they  should  also  avoid  the  worst.  The  prime 
desideratum  in  arithmetic  is  the  ability  to  work  accurately,  with 
reasonable  rapidity,  and  with  interest,  and  to  know  how  to  apply 
number  to  the  ordinary  affairs  of  life.  To  secure  accuracy  alone, 
to  secure  speed  alone,  to  have  arithmetic  a  play  spell  without 
accuracy  and  speed,  or  to  know  how  to  apply  number  to  life  in  a 
slovenly  way, — these  are  extremes  that  should  be  avoided  at  any 
cost,  including  the  tempting  cost  of  sensationalism.  It  is  the 


Method  25 

mission  of  the  training  school  or  college  to  make  the  earnest, 
well-balanced  teacher,  first  of  all.  With  this  duty  goes  the 
laudable  one  of  reasonable  experiment,  of  trying  out  sugges- 
tions from  whatever  source;  but  normal  schools  and  teachers 
colleges  must  at  all  hazards  guard  against  having  it  appear  that 
an  experiment  is  an  accomplished  result,  or  of  sacrificing  our 
children  in  unnecessary  quasi-clinical  work  that  is  doomed  to 
failure.  In  this  connection  one  of  the  resolutions  adopted  by 
the  National  Education  Association  in  1908  may  be  read  with 
profit  as  voicing  the  sentiment  of  that  s^ner  element  in  education 
that  is,  after  all,  the  strength  of  our  profession: 

"  We  recommend  the  subordination  of  highly  diversified  and 
overburdened  courses  of  study  in  the  grades  to  a  thorough  drill 
in  essential  subjects;  and  the  sacrifice  of  quantity  to  an  improve- 
ment in  the  quality  of  instruction.  The  complaints  of  business 
men  that  pupils  from  the  schools  are  inaccurate  in  results  an& 
careless  of  details  is  a  criticism  that  should  be  removed.  The 
principles  of  sound  and  accurate  training  are  as  fixed  as  natural 
laws  and  should  be  insistently  followed.  Ill-considered  experi- 
ments and  indiscriminate  methodising  should  be  abandoned,  and 
attention  devoted  to  the  persevering  and  continuous  drill  necessary 
for  accurate  and  efficient  training;  and  we  hold  that  no  course  of 
study  in  any  public  school  should  be  so  advanced  or  so  rigid  as 
to  prevent  instruction  to  any  student,  who  may  need  it,  in  the 
essential  and  practical  parts  of  the  common  English  branches." 

BIBLIOGRAPHY:  The  author's  Teaching  of  Elementary  Mathe- 
matics, pp.  71-97,  a  rather  extensive  discussion;  Seeley,  Grube's 
Method,  New  York,  1888;  Soldan,  Grube's  Method,  Chicago, 
1878;  C.  A.  McMurry,  Special  Method  in  Arithmetic,  New 
York,  1905 ;  McLellan  and  Dewey,  The  Psychology  of  Number, 
New  York,  1895;  Young,  pp.  53-150;  Hornbrook,  Laboratory 
Method  of  Teaching  Mathematics,  New  York,  1895 ;  Perry,  The 
Teaching  of  Mathematics,  London,  1902 ;  Suzzallo,  loc.  cit.; 
Ballard,  The  Teaching  of  Mathematics  in  London  Public  Ele- 
mentary Schools,  London,  1911. 


CHAPTER  VII 
MENTAL  OR  ORAL  ARITHMETIC 

The  objection  to  the  expression  "  Mental  Arithmetic  "  is  fully 
a  generation  old.  It  is  argued  that  written  arithmetic  is  quite  as 
mental  as  any  other  kind,  and  that  the  opposite  to  written  is  oral. 
As  to  this  there  can  be  no  argument,  but  the  word  "  mental  "  has 
so  long  been  used  to  apply  to  that  phase  of  arithmetic  that  is  not 
dependent  upon  written  help  that,  like  a  person's  proper  name,  it 
need  not  be  held  strictly  to  account  for  what  it  literally  signifies. 
The  expression  "  Mental  Arithmetic  "  is  therefore  employed  as 
well  as  "  Oral  Arithmetic  "  in  this  article  simply  because  it  is 
historical  and  of  well-understood  significance. 

What,  now,  are  the  relative  claims  of  written  and  mental  arith- 
metic? Historically,  the  mental  long  preceded  the  written,  but 
only  in  very  simple  problems,  chiefly  involving  counting  and  easy 
addition.  As  soon  as  the  writing  of  numbers  was  introduced, 
written  arithmetic  or  else  the  arithmetic  of  some  form  of  the 
abacus  became  practically  universal.  In  Japan  to-day  a  native 
shopkeeper  will  multiply  2  by  6  upon  the  soroban  (abacus),  and 
such  mechanical  aids  were  not  only  not  discarded  in  western 
Europe  until  the  sixteenth  century,  but  they  are  still  universal  in 
Russia.  About  the  beginning  of  the  last  century,  however,  mental 
arithmetic  underwent  a  great  revival,  largely  through  the  influence 
of  Pestalozzi  in  Europe  and  Warren  Colburn  in  this  country,  in 
each  case  as  a  protest  against  the  intellectual  sluggishness,  lack 
of  reasoning,  and  slowness  of  operation  of  the  old  written  arith- 
metic. For  a  long  time  the  mental  form  was  emphasized,  in 
America  doubtless  unduly  so,  and  was  naturally  followed  by  such 
a  reaction  that  it  lost  practically  all  of  its  standing.  The  question 
for  teachers  to-day  is  this,  what  are  the  fair  claims  of  these 
two  phases  of  the  subject  upon  the  time  and  energy  of  pupil  and 
teacher  ? 

There  are  two  points  of  view  in  the  matter,  the  practical  and 

26 


Mental  or  Oral  Arithmetic  27 

the  educational  or  psychological,  and  fortunately  they  seem  to 
lead  to  the  same  conclusion.  Practically  a  person  of  fair  intelli- 
gence should  not  need  a  pencil  and  paper  to  find  the  cost  of  6 
articles  at  2  cents  each,  or  of  5^4  yards  at  16  cents  a  yard.  The 
ordinary  purchase  of  household  supplies  requires  a  practical  ability 
in  the  mental  arithmetic  of  daily  life,  and  this  ability  comes  to  the 
mind  only  through  repeated  exercise.  As  will  be  seen  later,  it  is 
a  fair  inference  from  statistical  investigations  that  a  person  may 
be  rapid  and  accurate  in  written  work  but  slow  and  uncertain  in 
oral  solutions.  Therefore,  it  will  not  do,  from  the  practical  stand- 
point, to  drill  children  only  in  written  arithmetic  if  we  expect 
them  to  be  reasonably  ready  in  purely  mental  work.  On  psycho- 
logical grounds,  too,  the  neglect  of  mental  arithmetic  is  unwise. 
It  is  a  familiar  law  that  the  memory  is  stronger  on  a  fact  that  is 
known  in  several  ways  (a  convenient  phrase,  if  not  scientific) 
than  on  a  fact  that  is  known  in  only  one  way.  A  man  who  knows 
a  foreign  word  only  through  the  eye  may  forget  it  rather  easily, 
but  if  his  tongue  has  been  taught  to  pronounce  it,  even  though  he 
be  deaf,  he  can  the  more  readily  recall  it.  If  in  addition  to  this 
his  ear  has  often  heard  it  he  is  the  more  strongly  fortified,  and  if 
he  has  also  often  written  it,  by  pen  or  by  typewriter,  there  is 
this  further  chain  that  holds  it  to  the  memory.  In  other  words, 
the  greater  number  of  stimuli  that  we  can  bring  to  bear,  the 
more  certain  the  reaction.  Now  arithmetic  furnishes  merely  a 
special  case  of  this  general  law.  If  a  child  could  simply  see 
9  X  8  =  72  often  enough  he  would  come  to  be  able  to  write  it 
in  due  time,  even  if  he  did  not  know  the  meaning.  If  in  addition 
to  this  he  knows  the  meaning  of  these  symbols  and  recalls  having 
taken  9  bundles  of  8  sticks  each  and  finding  that  he.  had  72  sticks, 
then  the  impression  on  the  brain  is  the  more  lasting.  If,  further- 
more, he  has  been  trained  to  say  "  nine  times  eight  are  seventy- 
two  "  repeatedly,  the  impression  is  still  stronger,  and  if  he  has 
repeatedly  heard  this  expression  (and  here  is  one  of  the  advan- 
tages of  class  recitation)  he  has  then  a  still  further  mental  grip 
upon  the  fact.  In  other  words,  mental  arithmetic  in  the  form  of 
rapid  oral  work,  with  both  individual  and  class  recitation,  is  a 
valuable  aid,  psychologically,  to  the  retention  of  number  facts. 

There  is,  however,  a  danger  to  be  recognized.     It  is  asserted 
that  a  child  tires  more  quickly  of  abstract  work  than  of  genuine 


28  The  Teaching  of  Arithmetic 

concrete  problems;  problems,  that  is,  that  are  not  manifestly 
"  made  up  "  but  that  represent  some  of  his  actual  quantitative 
experiences.  Whether  he  really  tires  more  quickly  of  the  abstract 
than  of  the  concrete  is  by  no  means  certain ;  for  he  seems  to  have 
more  interest  in  the  former  than  in  the  latter,  probably  from  the 
added  difficulty  that  the  concrete  problem  presents  in  requiring 
him  to  know  what  operations  he  must  perform.  At  any  rate  he 
tires  of  both,  as  he  does  of  any  other  intellectual  exercise.  It 
therefore  follows  that  if  five  minutes  of  mental  work  produce  a 
certain  efficiency,  thirty  minutes  will  by  no  means  produce  six 
times  that  efficiency.  If,  now,  this  mental  work  is  valuable,  how 
much  time  and  energy  should  be  allotted  to  it?  Possibly  we 
shall  have  a  statistical  reply  to  this  question  sometime,  although 
it  will  be  a  sorry  day  for  good  teaching  if  we  should  ever  accept 
such  a  reply  as  final,  any  more  than  we  should  accept  the  crude 
statistics  of  the  health  department  as  determining  the  prescription 
our  physician  gives  us  for  indigestion.  The  statistics  may  help 
us,  but  they  can  never  control  us.  But  in  absence  of  even  their 
assistance,  what  shall  we  give  as  an  empirical  answer?  It  seems 
to  be  the  experience  of  teachers  generally  that  a  little  mental  work, 
rapid,  spirited,  perhaps  with  some  healthy,  generous  rivalry  to 
add  spice  to  the  exercise,  should  form  part  of  every  recitation 
throughout  the  course  in  arithmetic.  There  will  often  be  excep- 
tions, but  in  general  it  is  a  pretty  good  rule  to  devote  from  three 
to  five  minutes  daily,  and  sometimes  much  more  time,  to  this  kind 
of  work.  In  this  way  a  child  never  gets  out  of  practice,  save 
during  the  summer  holidays,  and  the  practical  and  psychological 
benefits  can  hardly  be  estimated. 

What  should  be  the  nature  of  this  mental  work?  On  the 
applied  side  there  is  no  better  test  for  the  teacher's  ability  to 
adapt  herself  to  her  environment,  educationally,  than  this,  for  the 
answer  varies  with  the  school  year,  the  locality,  the  related  sub- 
jects in  the  course,  and  with  many  other  factors.  In  general, 
however,  it  may  be  said  that  mental  arithmetic  offers  the  best 
means  for  correlating  the  subject  with  the  pupil's  other  work, 
both  within  and  without  the  school.  To  limit  it  to  this  field, 
however,  would  be  an  evident  mistake,  the  work  with  abstract 
number  demanding  the  major  part  of  the  time  assigned  to  this 
feature.  To  acquire  perfect  mechanical  reaction  to  a  given 


Mental  or  Oral  Arithmetic  29 

stimulus  much  exercise  is  required,  and  for  a  child  to  think  72 
when  stimulated  by  the  ideas  9X8  and  8X9  demands  repeated 
practice,  not  merely  in  relatively  few  applications  but  in  a  multi- 
tude of  questions  involving  abstract  numbers.  Nor  is  this  practice 
any  more  irksome  than  is  the  solution  of  applied  problems,  as  any 
teacher  knows.  It  was  almost  exclusively  by  this  abstract  work 
that  Pestalozzi  developed  calculators  of  such  ability  with  con- 
crete problems  as  astonished  those  who  visited  his  school,  al- 
though, if  we  may  place  confidence  in  the  results  of  Dr.  Stone's 
recent  investigations,  ability  in  either  of  these  lines  does  not 
necessarily  imply  ability  in  the  other. 

In  conclusion,  we  have  two  lines  of  work  in  mental  arithmetic ; 
(i)  the  concrete,  in  which  the  teacher  has  an  excellent  oppor- 
tunity for  correlation,  for  local  color,  and  for  stimulating  the 
interest  in  the  uses  of  arithmetic;  (2)  the  abstract,  in  which  the 
text-book  may  be  trusted  to  furnish  a  considerable  part  of  the 
material.  Each  must  be  cultivated,  and  ability  in  one  does  not 
necessarily  mean  a  corresponding  standard  of  ability  in  the  other, 
although  a  failure  in  the  abstract  line  must  lead  to  a  failure  in 
the  concrete.  One  leads  to  the  acquisition  of  number-facts,  the 
other  to  the  ability  to  rationally  use  these  facts  in  applied 
problems. 

As  a  practical  question  for  the  teacher,  how  is  the  material 
for  this  oral  work  to  be  found  ?  The  answer  is  evident ;  it  must 
be  found  exactly  as  we  find  material  in  geography,  in  history, 
and  in  written  arithmetic, — from  a  text-book.  No  teacher  can 
make  up  on  the  spur  of  the  moment  all  of  the  oral  examples 
necessary,  and  arrange  them  properly,  and  cover  all  of  the  impor- 
tant phases  of  drill  work.  Either,  then,  a  book  must  be  used  that 
supplies  both  the  oral  and  the  written  work,  or  else  two  books 
must  be  used,  one  for  the  oral  and  one  for  the  written.  In  either 
case  there  should  be  a  good  supply  of  oral  problems,  to  be  supple- 
mented by  the  teacher  with  such  local  problems  and  such  correla- 
tion with  other  work  as  may  be  advisable. 

BIBLIOGRAPHY:  The  author's  Teaching  of  Elementary  Mathe- 
matics, p.  117;  Handbook  to  Arithmetics,  Boston,  1904,  p.  6; 
Young,  p.  230;  C.  W.  Stone,  Arithmetical  Abilities;  Wentworth- 
Smith,  Oral  Arithmetic,  Boston,  1909. 


CHAPTER  VIII 
WRITTEN  ARITHMETIC 

What  has  been  said  of  mental  arithmetic  naturally  leads  to 
some  question  as  to  the  nature  of  the  written  work.  What  shall 
this  be?  If  the  difference  in  longitude  between  two  ships  (since- 
standard  time  by  one  system  or  another  is  now  coming  to  be 
universal  on  land)  is  33°  45',  how  shall  a  pupil  find  the  difference 
in  time?  Here  are  a  few  possibilities: 

(I)  (2) 

2  hr.     15  min  2  hr.     15  min. 

15)33°  45'  I5°)33°  45' 

30  30 


3°  45'  =  225'  3°  45' =  225' 

15  15 


75  75 

75  75 


o(3)  (4) 

33    45'  33    45' 

60 


1980 
45 


30 


15)2025(135  min.  =  2  hr.  15  min. 


52 

45 

75 

(5) 

33°  45'  =  33M° 
33#XY,.  hr.  =  2#  hr. 

30 


Written  Arithmetic  31 

Numerous  other  forms  could  be  suggested,  but  these  will  suf- 
fice for  our  purposes.  Which  of  these  should  be  preferred?  In 
general,  should  we  recommend  the  form  that  gives  us  the  result 
most  quickly,  or  some  other  that  may  show  clearer  reasoning? 
In  Nos.  i  and  4  the  forms  indicate  that  we  divide  degrees  by  an 
abstract  number  and  get  hours  instead  of  degrees;  in  No.  2  we 
seem  to  divide  degrees  by  degrees  and  get  hours  instead  of  an 
abstract  number ;  in  No.  3  we  seem  to  multiply  degrees  and  get 
an  abstract  number,  and  to  divide  one  abstract  number  by  an- 
other and  get  concrete  time  in  the  quotient;  in  No.  5  we  omit 
part  of  work  of  reduction  but  otherwise  the  solution  is  a  truth- 
ful one,  with  none  of  the  errors  of  reasoning  of  the  rest.  .This 
problem  has  been  selected  as  the  first  for  consideration  because 
it  opens  at  once  such  a  wide  range  of  possibilities  of  form,  but 
essentially  the  same  question  repeatedly  occurs  from  the  very 
first  grade  through  the  pupil's  school  life.  If  i  yard  of  cloth 
costs  I5c.,  what  wiH  6  yards  cost? — is  a  simpler  question  to 
consider.  Here  we  have  these  possibilities,  among  others: 

(i)  (2)  (3) 

IS  15  i5c. 

666 

90  9oc.  pec. 

(4)     6X15  =  90  (5)     6Xi5  =  9oc. 

(6)     6Xi5c.  =  goc.  (7)  15  X   6  =  cpc. 

Out  of  all  these,  which  shall  a  child  use  in  writing  a  solution? 

In  each  of  these  problems  the  fundamental  question  is  the 
same:  Shall  written  work  be  considered  from  the  standpoint  of 
the  answer  only,  as  a  business  man  would  be  inclined  to  do,  or 
from  the  standpoint  of  the  logic  of  the  school,  the  often  non- 
practical  school? 

The  answer  to  such  a  question  ought  not  to  be  dogmatic  to 
the  extent  of  saying  that  any  one  form  is  always  the  best,  al- 
though it  may  say  that  those  forms  that  are  untrue  in  statement 
are  always  bad.  That  is  to  say,  sometimes  it  is  better  to  write 

15)33 45  15 

2     15)225  6 

15 
2hr.  15  min.  90 

At  other  times  the  step  form,  with  the  denomination  accurately 


32  The  Teaching  of  Arithmetic 

set  forth,  is  better.  We  need  to  distinguish  between  two  lines  of 
work,  equally  important;  the  one  relates  to  accuracy  and  speed 
in  operation,  the  getting  of  an  answer  as  a  business  man  would, 
with  no  circumlocution  and  no  superfluous  symbols  or  opera- 
tions ;  this  is  the  mechanical  part  of  the  problem  and  there  must 
be  abundant  exercise  on  this  side.  Then  there  is  the  "equally  im- 
portant side  of  the  reasoning,  explaining  why  the  mechanical 
work  is  performed  as  it  is,  why  we  multiply  instead  of  divide,  and 
how  we  know  that  the  result  is  hours  instead  of  degrees,  or  cents 
instead  of  yards  of  cloth.  Here  the  step  form  of  analysis  may 
be  depended  upon  to  show  the  pupil's  line  of  reasoning.  These 
two  lines  of  written  work  are,  therefore,  legitimate.  What,  then, 
is  illegitimate  in  written  work,  and  what  are  the  dangers  to  be 
guarded  against  in  that  which  we  do  adopt?  As  to  the  first,  it 
may  be  laid  down  as  axiomatic  that  a  form  that  states  or  seems 
to  state  a  falsehood  is  illegitimate.  That  is,  30°  -r-  15  =  2  hr. 
is  a  false  statement;  it  is  not  even  excusable  on  the  score  of 
brevity,  since  30  X  a/15  nr-  =  2  nr-  is  as  brief,  is  true,  and  is  as 
easily  explained  as  any  form.  So  6  X  15  =  goc.  is  a  false  state- 
ment and  should  not  be  tolerated,  although  6  X  15  =  90,  or 
6  X  I5c.  =  Qoc.,  is  legitimate.  And  as  to  the  dangers  against 
which  to  guard,  the  following  advice  may  be  given :  ( I )  To 
require  that  every  applied  problem  should  be  solved  in  steps  is 
to  encourage  arithmetical  dawdling;  the  pupils  should  continu- 
ally be  exercised  in  rapid  solution,  the  correct  answer  speedily 
ascertained,  as  a  business  man  would  get  it,  being  the  aim.  A 
pupil  who  lets  his  mind  continually  dwell  upon  dollar  signs  and 
well  written  steps  cannot  help  but  drop  away  from  strict  atten- 
tion to  rapidity  and  accuracy  of  calculation.  (2)  To  split  hairs 
on  questions  of  such  forms  as  9  X  I5c.  or  150  X  9  is  to  get  away 
from  the  essential  point ;  we  must  recognize  the  fact  that  there  is 
good  authority  for  both,  although  the  former,  writing  the  sym- 
bols in  the  order  they  are  read,  is  coming  into  rather  general  use 
in  America.  The  great  question  is  to  see,  in  these  analyses,  that 
the  thought  is  clear,  and  that  a  pupil  is  not  thinking  in  a  hazy 
way  of  "15  cents  times  9."  (3)  To  require  no  analyses  of  the 
applied  problems  is  an  extreme  that  is  about  as  bad  as  to  require 
them  for  all,  and  perhaps  worse.  It  is  quite  sure  to  result  in 
looseness  of  reasoning  that  makes  correct  results  mere  matters 


Written  Arithmetic  33 

of  luck.  (4)  To  require  some  particular  form  of  analysis,  only 
to  meet  the  idiosyncrasy  of  the  teacher,  is  also  a  danger  against 
which  we  need  to  be  on  our  guard.  For  example,  always  to 
require  a  solution  stated  in  one  step,  if  possible,  is  a  hobby  that 
some  like  to  ride,  because  it  seems  to  demand  continued  thought, 
although  it  is  entirely  foreign  to  the  plan  that  a  common-sense 
business  man  would  adopt,  and  is  not  the  form  of  reasoning 
that  we  commonly  take  in  mathematics.  So  to  require  that  a 
child  should  always  take  some  unitary  form  of  analysis,  finding 
in  every  case  what  one  thing  costs,  may  be  the  means  of  check- 
ing the  originality  and  dampening  the  ardor  of  some  very 
promising  pupil. 

In  general,  therefore,  the  teacher  should  see  to  it  that  there  is 
a  reasonable  amount  of  rapid,  accurate  solution,  the  "  answer  " 
being  the  paramount  object.  He  should  also  see  that  there  is  a 
reasonable  amount  of  written  analysis,  accurately  stated,  prefer- 
ably in  the  convenient  and  terse  form  of  steps,  but  not  limited  in 
any  notional  way  that  would  destroy  originality  or  make  a  solu- 
tion unnecessarily  long. 

In  the  marking  of  papers  it  should  be  born  in  mind  that  there 
is  only  one  test  for  a  question  involving  a  single  operation. 
Either  the  answer  is  right  or  it  is  wrong.  If  the  problems 
require  some  interpretation,  a  teacher  may  properly  mark  both 
for  operations  and  for  method;  that  is,  a  pupil  may  perform 
his  operations  correctly,  but  may  have  misinterpreted  the  mean- 
ing of  the  problem.  In  that  case  some  credit  may  properly  be 
given  for  the  correct  operation.  In  general,  however,  papers  in 
arithmetic  should  be  marked,  as  they  are  in  business,  largely  by 
the  accuracy  of  the  result.  In  any  single  operation  the  work  is 
right  or  it  is  ivrong.  A  business  man  will  not  excuse  a  book- 
keeper who  writes  $9250.75  instead  of  $90,250.75.  Ony  a  zero 
is  missing,  but  it  means  a  difference  of  over  $80,000.  If  the 
result  is  wrong,  the  paper  is  wrong.  The  converse  of  this  state- 
ment is  not  true,  for  the  result  may  be  right,  and  yet  the  paper 
may  be  justly  criticised  for  its  slovenly  appearance  and  the  inac- 
curacy of  the  forms  used.  Where  a  time  limit  has  been  set,  and 
a  class  has  been  given  twenty  minutes  to  solve  as  many  problems 
as  possible,  teachers  must  use  their  judgment  as  to  marking 
pupils  who  are  naturally  slow.  If  their  work  is  accurate,  and 


34  The  Teaching  of  Arithmetic 

they  have  done   a   reasonable  number  of  examples,   they   are 
entitled  to  credit  and  should  receive  commendation. 

BIBLIOGRAPHY:  The  author's  Teaching  of  Elementary  Mathe- 
matics, pp.  121-129;  Handbook  to  Arithmetics,  p.  8;  Practical 
Arithmetic,  Boston,  1906,  pp.  115,  159;  Wentworth-Smith,  Com- 
plete Arithmetic,  and  Arithmetic,  Book  II,  Boston,  1909  and 
1911,  p.  191. 


CHAPTER  IX 
CHILDREN'S  ANALYSES 

The  questions  of  mental  and  written  arithmetic  lead  naturally 
to  that  of  the  analyses  to  be  expected  on  the  part  of  children. 
What  is  their  object,  what  should  be  .their  nature?  How  exten- 
sively should  they  be  required? 

As  to  the  first,  the  only  defensible  object  would  seem  to  be 
that  through  these  analyses  a  child  makes  it  clear  that  he  under- 
stands a  particular  problem  or  operation.  That  he  acquires  a 
habit  of  formal  statement  that  is  helpful  in  other  lines  of  work, 
or  that  his  memory  is  strengthened  by  learning  set  forms  of 
analysis,  has  been  too  often  disproved  to  require  argument. 
To  the  extent  that  this  analysis  is  really  an  explanation  of  his 
process  there  is  an  unquestionable  advantage,  since  it  enables  a 
teacher  to  commend  or  improve  the  pupil's  work.  But  how 
often  is  this  the  case?  Indeed,  how  often  should  it  be  expected 
to  be  the  case?  Is  it  not  the  general  experience  that  pupils  too 
often  memorize  their  analyses,  and  that  teachers  commend  glib 
repetitions  of  their  own  words  or  those  of  the  text-book,  the 
matter  being  so  imperfectly  comprehended  by  the  child  that  he 
is  able  to  bear  no  questioning? 

To  take  a  concrete  case,  we  occasionally  hear  some  teacher 
say  that  not  a  child  in  the  class  can  explain  why,  in  dividing  by 
a  fraction,  he  inverts  the  divisor  and  multiplies.  But  why  should 
he  explain  it?  And  if  he  does,  will  he  do  any  more  than  repeat 
in  a  perfunctory  way  the  analysis  he  learned  from  the  book  or 
the  teacher?  It  took  the  world  thousands  and  thousands  of 
years  to  learn  this  process.  It  was  a  thousand  years  after 
Euclid  made  his  great  geometry  before  it  was  used,  and  nearly 
another  thousand  years  elapsed  before  it  appeared  in  a  printed 
book.  This  means  that  maturity  of  mind  was  required  to  de- 
velop such  a  process,  and  still  greater  maturity  was  needed  to 
embody  it  in  a  text-book. 

35 


36  The  Teaching  of  Arithmetic 

But  does  this  mean  that  no  explanations  are  to  be  given  or 
required?  By  no  means.  A  child  should  know  this  process  of 
dividing,  and  he  should  learn  it  by  a  teacher's  questioning;  he 
should  thereby  know  that  it  is  reasonable,  and  he  should  feel  that 
for  the  time  he  understands  why  he  proceeds  in  this  manner. 
For  that  occasion  he  may  be  questioned  as  to  all  this,  but  that 
he  should  long  remember  the  "  why  "  of  it  all,  or  that  he  should 
be  able,  at  any  time  that  some  teacher  or  supervisor  thinks  fit, 
to  give  a  lucid  explanation  of  such  a  mature  process  is  as  un- 
natural as  it  is  unscientific. 

So  it  is  with  the  fundamental  operations  in  general.  There 
is  no  good  reason  why  a  child  should  remember  for  any  consid- 
erable time  an  explanation  for  multiplying  one  integer  by  an- 
other; it  is  sufficient  that  he  learned  the  operation  as  a  rational 
one,  and  that  he  can  perform  it  quickly  and  accurately  as  we  can 
or  as  any  business  man  does.  If  he  does  give  an  explanation  it 
will  usually  be  found  to  be  merely  a  parrot-like  repetition  of  the 
teacher's  or  the  text-book's  words,  without  any  apparent  mental 
content. 

In  the  matter  of  the  applied  problems  the  case  is  different. 
So  long  as  a  pupil  does  not  blindly  recite  formal  analyses,  there 
may  be  a  good  deal  of  value  in  his  explanations.  If  allowed  to 
state  his  reasons  in  his  own  language,  with  limitations  as  to 
tolerable  English,  he  may  acquire  a  habit  of  succinct  and  logical 
statement  that  will  help  him  in  many  other  lines  of  expression. 
This  affords,  moreover,  a  very  good  opportunity  for  the  teacher's 
commendation  and  advice, — criticism  in  the  best  sense  of  the 
term,  the  word  too  often  being  employed  to  signify  mere  fault- 
finding. 

My  colleague,  Professor  Suzzallo,  has  properly  called  atten- 
tion to  the  fact  that  the  problem  of  teaching  children  to  reason 
in  arithmetic  is  twofold:  (i)  "  It  is  a  matter  of  the  ability  to  use 
language;  (2)  It  is  a  matter  of  good  thinking."  The  former 
has  been  confused  with  the  latter  by  most  teachers,  it  being  felt 
that  if  the  child  repeated  the  book  language  of  reasoning  he  was 
satisfying  the  demand  for  honest  thinking.  Genuine  training 
in  reasoning  is  not  this,  however;  it  is  a  carefully  thought  out 
process,  beginning  with  problems  involving  only  a  single  step, 
and  leading  gradually  to  those  involving  two  steps.  This  is  a 


Children's  Analyses  37 

reasonable  limit  of  primary  work,  problems  involving  three  steps 
being  rather  matters  for  the  intermediate  grades. 

In  all  this  work  it  should  be  borne  in  mind  that  there  are 
three  things  that  are  properly  demanded  at  one  time  or  another, 
but  not  necessarily  for  each  problem  that  is  solved.  These 
three  are:  (i)  to  work  rapidly  and  accurately;  that  is,  to  take 
the  shortest  road  to  the  answer,  and  to  be  certain  the  answer 
is  correct;  (2)  to  put  neatly  on  paper  not  merely  the  operation 
but  a  brief  explanation;  (3)  to  give  a  brief  analysis  or  oral 
explanation. 

For  example,  if  5  yd.  of  cloth  cost  $2.10,  how  much  will 
12  yd.  cost? 

(i)   The  number  work: 

.42  $.42 

12  X  $2.IO  12 


5  84 

42 
$5.04 

(2)  The  written  explanation: 

5  yd.  cost  $2.10.  i  yd.  costs  1/5  of  $2.10. 
12  yd.  cost  12  X  VB  of  $2.10,  or  $5.04. 

(3)  The  oral  analysis: 

Since  5  yd.  cost  $2.10,  i  yd.  will  cost  Vs  of  $2.10,  and  12  yd. 
will  cost  12  X  1/5  of  $2.10,  or  $5.04. 

Teachers  will  not  need  to  call  for  all  this  work  with  every 
example.  Sometimes  it  will  be  necessary  to  emphasize  (2)  and 
sometimes  (3),  but  the  important  thing  is  that  (i)  should  be 
quickly  and  neatly,  and,  above  all,  accurately  done.  One  of  the 
best  ways  to  secure  this  accuracy  and  to  avoid  absurd  answers 
is  to  estimate  the  result  in  advance.  The  pupil  should  write 
down  this  estimate  and  compare  it  with  the  answer,  and  if  there 
is  a  great  difference  look  over  his  work  again. 

For  example,  if  5  yd.  cost  $2.10  we  know  12  yd.  will  cost 
nearly  2l/2  times  as  much,  or  somewhere  near  $5.  When  we 
solve,  if  we  find  such  a  result  as  $50.40,  we  see  at  once  that  there 
is  a  mistake,  probably  in  the  position  of  the  decimal  point.  The 
correct  result  is  $5.04. 


38  The  Teaching  of  Arithmetic 

As  an  example  of  written  work,  involving  both  the  computa- 
tion and  the  analysis,  the  following  may  be  considered: 

A  merchant  bought  800  yd.  of  linen  lawn  at  67^ c.  a  yard, 
and  sold  725  yd.  at  8oc.  a  yard,  and  the  rest  at  a  bargain  sale 
at  65c.  a  yard.  Find  his  profit. 

$0.67^  $0.65  $725 

800  75  .80 


400  325  $580.00 

536  455  48.75 


$540  $48.75  $628.75 

540. 


$88.75 
The  written  analysis  is  as  follows: 

1.  800  X  $0.67^  =  $540,  the  cost. 

2.  800  yd.  —  725  yd.  =  75  yd.,  sold  at  bargain  sale. 

3.  75  X  $0.65  =  $48.75,  received  at  bargain  sale. 

4.  725  X  $0.80  =  $580,  received  at  regular  sale. 

5.  $580  +  $48.75  =  $628.75,  total  receipts. 

6.  $628.75  -  $540  =  $88.75,  Profit. 

Of  course  this  solution  could  easily  be  shortened,  but  for  be- 
ginners it  is  as  well  not  to  attempt  too  much  brevity. 

In  conclusion  it  may  be  said  that  set  forms  of  analysis  seem 
to  be  rather  more  harmful  than  helpful  to  children,  but  that 
explanations  in  their  own  language  may  be  the  means  of  acquir- 
ing valuable  habits  and  of  offering  to  the  teacher  the  opportunity 
for  helpful  suggestions.  To  acquire  this  power  the  child  must 
be  initiated  gradually,  first  in  simple  examples  in  one-step  rea- 
soning, and  second  in  two-step  reasoning. 

BIBLIOGRAPHY  :  Young,  p.  205 ;  The  author's  Handbook  to 
Arithmetics,  p.  9;  Suzzallo,  loc.  cit. 


CHAPTER  X 
INTEREST  AND   EFFORT 

There  has  of  late  years  been  a  tendency  throughout  the  country 
to  make  arithmetic,  as  other  subjects,  more  interesting  to  children. 
What  the  real  motive  was  it  is  hard  to  say,  since  it  was  probably 
somewhat  subconscious.  Such  statistical  information  as  we  have 
shows  arithmetic  always  to  have  been  looked  upon  by  children 
as  one  of  the  most  interesting  subjects  of  the  course,  so  that  the 
reason  was  not  that  it  was  relatively  a  dull  study.  Possibly  the 
desire  was  that  the  work  of  the  teacher  should  become  easier 
through  increased  interest  on  the  part  of  the  pupils.  But  whatever 
the  reason  it  cannot  be  questioned  that,  other  things  being  always 
kept  equal,  there  is  a  great  gain  in  increasing  the  interest  in  any 
kind  of  work. 

There  is,  however,  a  general  danger  accompanying  this  effort 
to  increase  interest.  If  this  increase  means  that  the  subject  is  to 
become  anaemic,  if  it  is  not  to  require  the  same  serious  effort  to 
master  it  as  heretofore,  then  it  loses  a  considerable  part  of  the 
value  that  has  generally  been  assigned  to  it.  Moreover,  through 
this  same  cause  it  loses  a  considerable  part  of  the  very  interest 
that  was  expected  to  be  fostered.  Boys  and  girls  do  not  like  to 
wrestle  with  infants  or  with  infantile  subjects,  and  unless  a  study 
is  suitably  graded  as  to  difficulty  it  will  appeal  in  vain  to  the 
interest,  the  vigorous  attack,  and  the  responsive  mental  effort  of 
the  pupils. 

Our  lesson,  therefore,  is  that  we  should  do  all  in  our  power 
to  make  arithmetic  interesting  and  even  attractive  to  the  children, 
but  that  we  must  not  hope  to  attain  this  result  by  offering  a  sickly 
substitute  for  the  vigorous  subject  that  has  come  down  to  us. 
Unfortunately  we  have  not  been  free  from  this  fault  of  making 
our  arithmetic,  and  particularly  our  primary  arithmetic,  anaemic. 
Foreign  critics  frequently  comment  upon  this  failing,  and  claim 
with  good  reason  that  much  of  our  work  in  the  early  grades 

39 


4O  The  Teaching  of  Arithmetic 

lacks  vitality.  Certain  it  is  that  in  spite  of  many  points  of 
superiority  of  the  American  school  we  do  not  at  the  end  of  eight 
years  bring  our  children  as  far  as  European  experience  would 
seem  to  lead  us  to  hope. 

How  can  the  interest  in  the  applications  of  arithmetic  be 
aroused  and  maintained?  The  reply  has  already  been  made. 
They  must  be  real  if  they  pretend  to  be  so,  they  must  relate  when 
possible  to  the  child's  daily  environment,  and  they  must  reveal 
the  life  of  America  to-day  in  such  a  way  as  to  be  broadly  informa- 
tional as  well  as  mathematical.  This  can  be  accomplished  with 
no  less  demand  for  mental  power  than  was  required  by  the  obso- 
lete problems  of  our  old-style  books.  There  are,  however,  various 
other  channels  through  which  we  may  pass  to  reach  the  required 
end.  For  example,  there  are  the  number  games  for  children  of 
the  primary  grades,  games  that  have  an  interest  that  pleasantly 
conceals  the  mental  effort  required,  as  tennis  does  the  muscular 
effort,  but  that  accomplish  the  result  efficiently.  This  subject  is 
considered  later  in  this  work.  Then  there  are  the  problems  of 
heroic  effort,  then  of  mechanical  effort,  then  of  simple  building, 
and  later  of  our  national  resources.  These  resources,  correlating 
as  they  do  with  geography,  concern  our  supply  of  food,  and 
clothing,  and  home  comforts;  they  touch  our  mines,  our  farms, 
our  transportation,  and  the  great  industries  of  our  people. 

As  concrete  illustrations  of  this  type  of  problem  the  following 
relating  to  some  of  the  metals  produced  in  this  country  may  be 
considered : 

1.  When  700  Ib.  of  copper  was  worth  $73.50,  what  was  250 
Ib.  of  the  same  quality  worth? 

2.  A  dealer  sold  800  Ib.  of  nickel  at  55  ct.  a  pound,  thus  gain- 
ing 10%  on  the  cost.     How  much  did  it  cost  him? 

3.  If  a  dealer  buys  900  Ib.  of  zinc  at  $88  a  short  ton,  at  what 
rate  per  pound  must  he  sell  it  to  gain  50%? 

4.  In  one  year  silver  averaged  54.98  ct.  an  ounce,  which  was 
102.8%  of  the  average  for  the  preceding  year.     What  was  the 
average  then  ? 

5.  Silver  is  sold  by  the  troy  ounce.     This  is  what  per  cent 
of  the  avoirdupois  ounce  (7000  gr.  =  16  av.  oz.,  5760  gr.  = 
12  troy  oz.)  ? 


Interest  and  Effort  41 

6.  Quicksilver  (mercury)  is  sold  by  the  flask  of  76*^  Ib.     If 
a  dealer  buys  it  at  $38.25  a  flask,  at  how  much  a  pound  must 
he  sell  it  to  gain  20%  ? 

7.  If  a  short  ton  of  lead  is  worth  $96,  how  much  is  750  Ib. 
worth?    If  a  dealer  bought  it  at  this  rate  and  sold  it  at  8  ct.  a 
pound,  what  was  his  per  cent  of  profit? 

.The  interest  in  such  topics  is  measurably  greater  than  that  in 
equation  of  payments,  or  most  problems  in  compound  proportion, 
or  examples  in  the  Vermont  Rule  of  Partial  Payments  (a  subject 
that,  however,  naturally  has  its  place  in  the  curriculum  of  that 
state).  On  the  other  hand  the  effort  may  be  just  as  great  as 
we  wish  to  make  it.  It  is  only  a  matter  of  complicating  the 
problem  sufficiently,  and  using  numbers  and  combinations  of 
proper  difficulty,  to  make  a  modern  problem  about  the  coal  indus- 
try of  Pennsylvania,  or  the  silver  output  of  Colorado,  as  hard  as 
any  example  in  the  arithmetics  of  fifty  years  ago. 

We  have,  therefore,  the  following  points  that  seem  fair  con- 
clusions: (i)  It  is  possible  to  bring  our  arithmetic  work  to  a 
higher  plane  of  interest,  through  the  game  element  and  the  appli- 
cations ;  (2)  it  is  possible,  with  this,  to  keep  the  plane  of  effort  as 
high  as  we  wish;  (3)  with  the  increased  interest  must  necessarily 
come  an  increase  of  power  that  is  vital  to  the  improvement  of 
our  education. 

BIBLIOGRAPHY  :  In  the  matter  of  modern  problems  consult  the 
author's  arithmetics.  In  the  matter  of  mathematical  recreations 
consult  Ball,  Mathematical  Recreations,  London,  1896;  Schubert, 
Mathematical  Essays  and  Recreations,  Chicago,  1899. 


CHAPTER  XI 

IMPROVEMENTS  IN  THE  TECHNIQUE  OF 
ARITHMETIC 

Nothing  new  goes  into  arithmetic  without  a  protest,  and  so  for 
what  goes  out.  Nevertheless  there  has  been  an  evolution  here 
as  everywhere  else,  and  this  evolution  has  made  for  the  betterment 
of  the  subject.  To  take  a  concrete  illustration,  the  first  printed 
arithmetics  had  no  symbols  of  operation.  What  we  would  write 
as  "4X5"  was  then  written  "  4  times  5,"  with  the  natural 
variation  of  the  word  "  times  "  according  to  the  language  em- 
ployed. It  was  half  a  century  later,  and  after  the  symbols  -f-  and 
—  were  invented,  before  —  was  suggested,  and  some  eighty  years 
after  that  before  X  was  used,  and  a  long  time  after  that  before 
-r-  appeared  for  division.  It  was  several  generations  after  these 
were  first  used  before  they  came  into  our  school  arithmetics  for 
the  purposes  that  we  use  them  to-day,  and  always  with  strong 
protest  on  the  part  of  those  who  wish  to  "  let  well  enough  alone." 
It  was  argued  that  "4X5"  was  more  abstract  than  "  4  times  5," 
that  it  was  hard  because  of  the  symbolism,  and  that  it  took  arith- 
metic from  the  written  language  and  the  customs  of  the  common 
people  for  whom  it  was  of  greatest  use.  Invented  for  algebra, 
the  conservatives  said  that  all  the  symbols  ought  to  remain  there 
and  not  seek  to  enter  the  field  of  arithmetic.  This  struggle  of 
symbolism  seems  strange  to  us  to-day,  when  a  child  in  the  first 
grade  learns  at  least  half  a  dozen  signs  of  operation  and  relation, 
and  few  would  be  found  to  advocate  going  back  to  the  old  cus- 
tom. We  are,  however,  face  to  face  with  similar  questions  and 
many  of  the  very  ones  who  would  argue  to  keep  -4-  (a  symbol 
practically  unknown  save  in  England  and  America,  strange  as 
this  may  seem),  are  the  ones  who  protest  most  vehemently  against 
letting  x  stand  for  a  phrase  that  is  too  long  to  write  conveniently. 
But  the  question  is  the  same.  If  we  use  -r-  as  a  short-hand  way 
of  writing  "  divided  by,"  why  should  we  not  use  x  as  a  short- 

42 


Improvements  in  the  Technique  of  Arithmetic  43 

hand  way  of  writing  "  what  the  horse  cost,"  or  "  the  amount  due 
the  first  man,"  or  any  other  phrase  representing  a  quantity  to  be 
sought,  an  "  unknown  quantity "  ?  Here,  then,  is  one  of  the 
improvements  suggested  by  algebra  to  assist  us  in  reasoning  out 
the  solution  of  an  arithmetical  problem.  That  "  I.KXF  =  $3300, 
therefore  x  —  $3000  "  is  algebra  instead  of  arithmetic,  is  no  more 
true  than  that  "  4  X  5  "  is  algebra  while  "  4  times  5  "  is  arithmetic. 
The  symbol  X  was  used  only  in  algebra  instead  of  arithmetic  for 
a  century  or  so,  as  x  was  for  over  a  century  longer,  but  the 
employment  of  each  to  assist  in  arithmetic  does  not  make  "  a 
solution  by  algebra."  This  illustration  is  brought  forward  as  one 
of  the  most  prominent  at  the  present  time.  It  is  impressed  upon 
me  by  numerous  letters  asking  for  "  a  solution  by  arithmetic 
instead  of  by  algebra"  for  some  little  problem  that  is  made 
clearer  by  the  use  of  a  single  symbol  in  place  of  a  phrase  like 
"  the  number  of  bushels,"  or  "  the  cost  of  the  farm."  Teachers 
should  realize  that  they  hereby  show  an  ignorance  that  is  hardly 
pardonable  at  the  present  time,  and  that  such  improvements  in 
symbolism  are  a  part  of  the  natural  development  of  the  subject. 

Of  course  there  is  the  danger  of  overdoing  all  this.    This  has 
often  been  seen,  and  is  apparent  to-day.  For  example,  it  is  better 
to  train  a  child's  eye  to  see  that  4  and  5  are  9,  by 
putting  the  symbols  in  the  form  here  given,  the  form        4 
in  which  he  will  usually  meet  them  in  computation,         5 
than  to  train  it  with  the  symbolism  4  -f  5  =  9,  which       — 
he  rarely  sees  in  practical  work.     On  the  other  hand,        9 
to  neglect  the  latter  entirely  is  to  unfit  him  for  reading 
any  save  the  simplest  mathematics,  and  for  expressing  his  solu- 
tions in  the  condensed  step-form   necessary  to  allow  the   eye 
quickly  to  grasp  the  reasoning.     So  with  a  symbol  like  x;  we 
may  use  it  where  there  is  not  only  no  advantage  in  doing  so, 
but  a  positive  disadvantage.    It  is  the  part  of  the  text-book  and 
the  teacher  to  suggest  to  the  pupil  the  problems  in  which  it  should 
be  employed,  and  to  furnish  a  reasonable  amount  of  exercise  in 
the  subject. 

To  consider  a  still  more  definite  illustration,  take  this  problem : 
If  some  goods  are  sold  at  a  profit  of  \2.y2%  for  $1,012.50,  what 
did  they  cost?  Here  we  have  several  possible  lines  of  attack, 
among  which  are  the  following,  (i)  If  there  was  a  gain  of  i2l/2% 


44  The  Teaching  of  Arithmetic 

they  must  have  been  sold  for  112^  %  of  their  cost.  (But  where 
did  that  100  come  from?)  And  since  $102}^  is  H2y2%  of  the 
cost,  i%  of  the  cost  is  T}^  (But  how  are  we  to  read  such  a 
fraction?)  of  $1,012^,  and  100%  (But  why  do  we  wish  100%  ?) 
is  100  times  this.  Such  an  explanation,  while  it  could  be  learned 
and  recited,  would  be  subject  to  questions  like  those  inclosed  in 
the  parentheses,  and  really  would  have  very  little  reasoning  in  it. 

(2)  Let  1 00%  stand  for  the  cost.     (But  why  100%  instead 
of  200%  or  50%  or  some  other  number?)    Then  100%  +  i2l/2% 
will  stand  for  the  selling  price.    If  112^%  =  $1,012^  (But  an 
abstract    number,    1.12^/2,    cannot    equal    a    concrete    number, 
$1,012^%)  then  i%  =  $1,012^  -r-  112*6  (Why?  Why  not  mul- 
tiply? Why  not  divide  by  H2y2%  instead  of  112^?),  and  100% 
(why  not  find  500%?)  =  100  times  as  much,  or  $900.      This 
analysis  is  quite  as  superficial  and  unsatisfactory  as  the  first. 

(3)  Let  i  =  the  cost.     (But  why  not  let  2  equal  the  cost? 
and  why  take  I  instead  of  ioo%?)     The  rest  of  this  solution 
might  follow  the  lines  of  (i)  or  (2),  and  in  that  case  it  would  be 
equally  unsatisfactory.     Like  (2),  it  is  a  relic  of  the  old  and 
long-since  forgotten  method  of  "  False  Position,"  the  discarding 
of  which  caused  so  many  conservative  teachers  to  feel  that  arith- 
metic was  losing  its  mental  discipline.     So  it  would  be  possible 
to  give  a  number  of  other  plans  of  attack,  some  better  than 
the  above,  and  some  much  worse.     Consider,  however,  a  single 
other  plan. 

(4)  Let  c  =  the  cost,  a  very  natural  thing  to  do  since  it  is 
the  initial  letter.    Then  the  selling  price  is  c  +  .\2.y2c. 

But  c  +  ,i2J/2c  =  I.i2,y2c,  and  therefore  I.i2y2c  —  $1012.50. 
Dividing  equals  by  1.12^2,  £  =  $900. 

Now  we  are  not  troubled  here  about  any  100%,  or  i%,  or 
letting  i  equal  something  that  it  cannot  equal.  As  soon  as  we 
think  of  \2.y2%  as  the  same  as  0.12%%,  and  know  that  c  +  .I2l/2c 
=  i.i2jAc,  just  as  c  +  l/2c  =  iy2c,  the  case  is  exceedingly  simple. 

Here,  then,  is  one  of  the  places  in  which  the  technique  of 
arithmetic  has  been  greatly  improved  by  the  introduction  of  an 
easy  symbol  from  algebra,  as  it  was  years  ago  improved  by  the 
introduction  of  the  symbols  of  operation.  No  teacher  who  has 
ever  seriously  tried  to  use  this  symbol  has  ever  willingly  aban- 
doned it. 


Improvements  in  the  Technique  of  Arithmetic  45 

What  has  been  said  for  the  symbol  x  might  be  said  for  other 
symbols  if  we  needed  them.  It  is  of  no  particular  consequence 
that  we  use  4^  instead  of  V4  in  arithmetic,  because  we  do  not 
make  much  use  of  square  root  in  arithmetic,  but  if  we  did  the 
more  modern  symbol  would  deserve  a  place  for  its  influence  in 
later  work,  and  the  same  may  be  said  of  the  negative  number, 
the  parenthesis,  and  other  signs  of  algebra. 

Outside  of  symbolism,  however,  there  are  certain  improve- 
ments in  technique  that  demand  our  consideration.  One  relates 
to  subtraction,  and  in  order  to  have  it  satisfactorily  understood 
it  becomes  necessary  first  to  speak  of  it  historically.  .There 
have  been  several  successful  plans  for  teaching  subtraction ;  each 
has  endured  for  a  long  period,  and  some  of  them  are  in  use  to- 
day. Of  the  most  prominent  the  following  may  be  mentioned: 

(1)  The  complementary  method.    Instead  of  taking  8  from  13 
we  may  add  10  —  8  to  13  and  then  drop  10. 

This   depends  on  the  relation    13  —  8=13+         13         13 
(10  — 8)  — 10,  and  since  10  — 8  is  called  the          8          2 
complement  of  8  (to  the  number  10),  this  is 
known  as  the  complementary  method.     It  is          5          5 
very  old,  appearing  in  a  famous  Hindu  work 
of  the  I2th  century,  in  the  first  printed  arithmetic  (1478),  and  in 
numerous  other  text-books.    In  a  case  like  452  —  348  the  opera- 
tion would  be  as  follows :  2  +  2  =  4;  since  10  must 
452        be  dropped,  we  may  add   i  to  the  4  instead;  then 
348        5  —  5  =  0.  and  4  —  3=1.     That  is,  the  complemen- 
tary idea  need  be  used  only  when  the  minuend  is 
104        less  than  the  subtrahend.     This  example  is  actually 
taken  from  the  first  printed  arithmetic.     The  plan  is 
the  same  as  the  one  used  in  Pike's  famous  American  arithmetic 
a  century  ago,  and  some  teachers  still  employ  it.     It  is  essentially 
the  one  used  when  we  employ  co-logarithms  in  trigonometry. 

(2)  The  borrowing  and  repaying  plan,  a  name  that  we  may 
use  for  want  of  a  better  one.    This  may  be  illus- 
trated by  the  annexed   example,  taken   from  the        6354 
first     great     business     arithmetic     ever     printed        2978 
(Borghi's,  of  1484).    The  operation  is  as  follows: 

8  from  14,  6;  8  from  15,  7;  10  from  13,  3;  3  from        3376 
6,  3.    This  was  the  plan  advocated  most  often  by 


46  The  Teaching  of  Arithmetic 

the  early  printed  arithmetics,  and  the  expression  "  to  borrow  " 
was  a  common  one.  It  was  known  to  the  early  Hindu  arith- 
meticians and  in  Constantinople  before  the  invention  of  printing. 

(3)  Simple  borrowing, — to  continue  to  use  this  old  English 

expression.  In  this  method  the  computer  says,  "  7 
42  from  12,  5;  2  from  3,  I."  This  is  probably  the  most 
27  common  plan  in  use  to-day,  and  it  has  much  to  com- 

—  mend  it.     It  has   a  long    history,    appearing    in    the 
15       oldest  known  manuscript  on  arithmetic  in  English,  in 

Spain  in  the  I3th  century,  in  Italy  in  the  Middle  Ages, 
and  in  India  still  earlier.  It  has  not,  however,  been  nearly  as 
popular  as  the  second  plan,  although  more  generally  used  in  our 
country  to-day. 

(4)  The  left-to-right  method,  a  plan  that  has  had  a  long  history 
in  which  many  prominent  advocates  have  appeared.    It  is  more 
adapted  to  the  needs  of  a  professional  computer,  however,  than 
to  those  of  the  average  citizen,  and  may  therefore  be  dismissed 
with  this  mere  mention. 

(5)  The  addition,  "making  change,"  or  "Austrian"  method, 
a  vague  way  of  naming  several  related  plans.    The  efforts  made 
to  adopt  it  in  the  Austrian  schools,  and  the  consequent  notice 
taken  of  it  in  Germany,  have  been  the  cause  of  its  most  inap- 
propriate geographical  name.     As  a  definite  method  of 

17      subtraction  it  is  not  as  old  as  the  others,  although  it 

8  appears   in  the   i6th   century   in    Italy  and   has    had 

—  occasional  prominent  advocates  since.      It  consists  in 

9  finding  what  number  must  be  added  to  the  subtrahend 
to  make  the  minuend.    Thus  in  thinking  of  17  —  8  we 

think:  "8  and  9  are  17,"  writing  down  the  9.  If  the  numbers 
are  longer  we  may  proceed  in  either  of  two  ways, 

as  in  the  annexed  example.    Here  we  may  say :  "  8  423 

and  5  are  13 ;  4  and  7  are  n  ;  i  and  2  are  3."    Or  we  148 

may  say :  "  8  and  5  are  13 ;  5  and  7  are  12 ;  2  and  2  are      

4."    Of  these  two,  one  is  as  easily  explained  as  the  275 
other.    The  first  might  naturally  be  approached  thus : 

423  —  400  +  20  +  3  =  300  +  1 10  +  13 

148  —  IPO  -f-  40  +  8  =  IPO  +40+8 

200  +70+5 


Improvements  in  the  Technique  of  Arithmetic  47 

The  arrangement  for  the  second  would  be  as  follows: 
423  =  400  +  20  +  3 
148  —  IOQ  +  40  +  8 

Since  the  difference  will  be  the  same  if  we  add  the  same  number 
to  both  minuend  and  subtrahend,  we  will  add  10  to  each,  and 
then  100  to  each,  giving  the  following: 

400+  120  +  13 
200+  50+8 
200  +70+5 

Since  the  explanations  are  about  equal  in  difficulty,  we  may 
consider  the  sole  question  of  rapidity  in  practical  use,  both  as  to 
the  method  in  general  and  as  to  these  two  sub-methods  in 
particular. 

Is  the  general  plan  the  best  one?  On  the  side  of  advantages 
we  have:  (i)It  is  the  common  method  of  making  change.  If  I 
owe  $7.65  and  pay  $10  the  merchant  finds  the  change  and  I 
verify  his  work  by  saying :  "  65C.  and  5c.  are  7oc.,  and  3oc.  more 
makes  $i,  and  $2  more  makes  $10."  That  is,  we  find  the  differ- 
ence by  adding.  This  is  familiar  in  all  business  and  in  the 
school,  and  will  remain  so.  It  is,  therefore,  a  natural  plan  to 
use  in  all  subtraction.  (2)  It  avoids  the  necessity  for  learning 
a  separate  subtraction  table.  Everything  is  referred  at  once  to 
the  addition  table,  a  table  that  unfortunately  is  not  at  present 
known  any  too  well.  There  is,  therefore,  an  economy  of  time 
and  an  increased  efficiency  in  the  very  important  subject  of 
addition.  (3)  The  facts  of  addition  being  used  so  much  more 
often  than  those  of  subtraction,  there  is  naturally  an  increase 
in  speed  and  certainty  when  we  employ  the  addition  instead 
of  the  subtraction  table. 

On  the  other  side,  it  is  not  desirable  to  change  the  customs 
of  the  people  unless  there  is  a  decided  gain  by  so  doing. 
Parents  who  have  been  brought  up  on  one  plan,  and  who  help 
their  children  more  or  less  in  their  lessons,  do  not  easily  adapt 
themselves  to  a  new  one,  and  the  result  is  a  confusion  in  the 
child's  mind  that  is  most  unfortunate.  The  fair  question,  there- 
fore, is,  Is  it  worth  the  while  to  use  the  better  method? 

If  we  had  a  standard  method  known  by  all,  this  argument 
would  have  much  weight,  but  we  have  at  least  three  rather  com- 
mon ones,  with  several  variations,  already  in  use  in  this  country. 


48  The  Teaching  of  Arithmetic 

There  is,  therefore,  bound  to  be  more  or  less  confusion.  The 
only  thing  for  the  school  to  do,  then,  is  to  teach  the  method  that 
will  prove  in  the  long  run  to  be  the  most  rapid  and  accurate,  and 
this  seems  a  priori  to  be  the  "  Austrian,"  although  a  scientific 
investigation  of  the  matter,  on  sufficient  data,  is  desirable.  And 
of  the  two  or  three  sub-methods,  the  second  one  described  on 
page  46  seems  without  doubt  the  better. 

The  question  is,  it  should  be  repeated,  not  one  of  explanation, 
since  any  one  of  the  methods  is  easily  explained.  It  is  purely 
one  of  practical  utility,  in  which  the  teacher  should  divorce  him- 
self of  all  prejudice  in  favor  of  the  method  which  was  taught  to 
him  and  with  which  he  is  most  familiar,— a  somewhat  difficult 
thing  to  do  in  a  discussion  of  this  kind.  It  should  also  be  re- 
marked .  that  it  is  of  doubtful  policy  to  attempt  to  change  a 
method  that  a  child  already  knows  and  handles  easily,  inasmuch  as 
the  difference  in  value  is  not  great  enough  to  warrant  this  course. 
For  beginners,  however,  the  plan  suggested  is  probably  the  best. 

These  two  matters  of  improvement  in  technique  have  been 
mentioned,  not  as  so  important  in  themselves,  but  as  types  of 
various  changes  that  are  worthy  of  sympathetic  consideration. 
The  placing  of  the  quotient  over  the  dividend  in  long  division, 
instead  of  at  the  right  as  was  formerly  done,  so  as  to  make  more 
clear  the  position  of  the  decimal  point;  the  multiplying  of  both 
dividend  and  divisor,  in  the  division  of  decimals,  by  such  a  power 
of  10  as  shall  make  the  divisor  an  integer,  thus  avoiding  the  old 
difficulty  of  determining  the  number  of  integral  places  in  the 
quotient ;  the  giving  of  the  full  form  of  an  operation  before  the 
abridged  one,  in  explaining  the  process ;  the  writing  of  ratios  in 
the  familiar  form  of  fractions  in  the  first  steps  in  proportion; 
the  putting  of  the  unknown  quantity  first  instead  of  last  in  writ- 
ing a  proportion  and  the  using  of  x  to  represent  this  quantity, — 
these  are  some  of  the  other  improvements  in  technique  that  are 
getting  into  our  books  in  these  days  and  that  should  command 
our  interest.  To  make  these  more  clear,  the  space  not  allowing 
of  elaborate  discussion,  an  example  of  each  is  here  given.  Thus 
in  long  division  it  is  better  to  use  the  first  of  these  forms  than 
the  second,  for  the  reason  specified,  that  it  makes  clear  the 
position  of  the  decimal  point,— a  reason  that  holds  equally  true 
for  the  second  example  given  on  page  49: 


Improvements  in  the  Technique  of  Arithmetic  49 

14.24 


72)1025.28  72)1025.28(14.24 

72  72 


305  305 

288  288 


172  172 

144  144 


288  288 

288  288 


Similarly,  the  first  of  these  forms  is  better  than  the  second,  the 
problem  being  to  divide  102.528  by  0.72 : 

14^.4 

72)10252.8  0.72)102.528(142.4 

72  72 


305  .  305 

288  288 


172  172 

144  144 


288  288 

288  288 


In  the  early  stages  the  first  of  the  following  forms  is  better  than 
the  second: 

1424  1424 


72)102528  72)102528 

72000  =  looo  X  72  72 


30528  305 

28800  =  400  X  72  288 


1728  172 

1440=  20X72  144 


288  288 

288  =   4  X  72  288 


5o  The  Teaching  of  Arithmetic 

In  introducing  the  idea  of  proportion  it  is  better  to  begin  with 
known  symbols,  so  as  not  to  confuse  the  pupil  too  much.  Thus 
the  first  of  these  forms  is  better  than  the  second  in  the  early 
stages : 

x     39 

91    :    39    ::    7    :     (?) 

7     9i 

Indeed  it  is  always  better  to  use  =  than  : :,  and  the  latter  is  now 
happily  passing  into  oblivion  in  this  country. 

There  are  several  more  important  questions  for  the  future,  in 
relation  to  the  technique  of  computation,  but  these  can  be  men- 
tioned only  briefly.  The  first  relates  to  limits  of  accuracy  in  re- 
sults. This  does  not  mean  that  the  work  may  be  inaccurate, 
but  that  if  we  know  the  circumference  of  a  circle  only  to  two 
decimal  places  we  cannot  from  that  find  the  diameter  to  three  or 
more  decimal  places.  We  express  this  by  saying  that  the  result 
cannot  be  more  accurate  than  the  data.  Suppose,  for  example, 
we  have  made  several  measurements  of  the  circumference  of  a 
steel  shaft,  and  their  average  is  7.57  in.;  it  is  evidently  useless 
in  dividing  this  by  3.1416  to  carry  the  result  beyond  two  decimal 
places.  Required,  therefore,  to  divide  so  as  to  avoid  unneces- 
sary work.  This  is  accomplished  by  what  is  known  as  con- 
tracted division,  thus: 

241 

3.1416)7.57 
628 


i  29 
125 

4 
3 

Now  how  much  of  this  kind  of  work  are  the  schools  called  upon 
to  teach?  It  is  certainly  not  a  thing  that  many  people  will  need 
to  know,  and,  therefore,  it  is  properly  omitted  from  most  text- 
books to-day.  In  some  localities,  however,  it  might  very  pro- 
perly be  taught,  and  when  our  classes  in  physics  require  the 
mathematics  that  they  might  properly  demand,  it  may  become 


Improvements  in  the  Technique  of  Arithmetic  51 

necessary  to  teach  contracted  multiplication  and  division  in  the 
schools. 

Another  topic  is  logarithms.  In  all  engineering  computations 
this  labor-saving  device  is  used,  and  the  subject  is  easily  taught. 
What  shall  we  do  with  it  in  arithmetic?  So  far  as  grade  work 
is  concerned  there  is  nothing  to  be  done  at  present,  because  the 
demand  is  not  great.  But  we  can  hardly  say  what  the  future 
may  bring  forth,  and  their  use  may  become  much  more  common 
than  we  think  if  the  teachers  of  physics  and  the  advocates  of 
more  mathematics  in  manual  training  stir  up' the  question  of 
greater  facility  in  practical  calculation.  In  the  same  way  we 
may  yet  see  the  slide  rule  (a  simple  instrument  for  computing 
by  machinery)  find  a  place  in  business  or  technical  courses  in 
the  grades  of  the  elementary  school.  At  present  our  duty  points 
to  a  teacher's  knowledge  of  all  these  matters  of  technique,  and 
a  sympathetic  awaiting  of  the  time  when  some  or  all  may  de- 
mand more  serious  attention  on  the  part  of  the  school. 


CERTAIN  GREAT  PRINCIPLES  OF  TEACHING 
ARITHMETIC 

Before  considering  the  curriculum  in  arithmetic  it  is  well  to 
devote  a  little  attention  to  certain  great  principles  that  teachers 
have  as  a  whole  agreed  upon,  in  theory  if  not  in  practice.  Some 
of  these  have  already  been  discussed  in  this  article;  others  will 
strike  the  reader  as  rather  trite,  which  simply  means  that  they  are 
generally  accepted ;  and  others  will  not  appeal  to  all.  They  will, 
however,  be  found  to  be  suggestive  of  the  thought  of  leaders  at 
the  present  time,  and  they  may,  though  stated  dogmatically, 
form  the  basis  for  profitable  discussions  by  teachers. 

1.  Arithmetic  is  taught  both  for  its  usefulness  in  daily  life 
and  for  the  training  that  it  gives  the  mind  in  reasoning,  in  habits 
of  application,  and  in  exactness  of  statement. 

2.  Most  of  the  mental  discipline  of  arithmetic  can  be  secured 
from  those  portions  that  may  be  called  practical,  and  therefore 
the  practical  side  of  arithmetic  may  safely  be  emphasized. 

3.  But  in  emphasizing  this  practical  side  we  need  to  offer  a 
large  amount  of  abstract  work  as  well  as  concrete  problems,  skill 
in  either  not  necessarily  signifying  skill  in  the  other. 

4.  In  the  concrete  problems,  whatever  pretends  to  be  genuine, 
to  represent  practical  questions  of  American  life,  should  be  so, 
all  obsolete  business  problems  being  replaced  by  modern  ques- 
tions.    In  particular,  the  daily  industries  of  our  people  should 
be  drawn  upon  to  the  making  of  arithmetic  interesting,  informa- 
tional, practical. 

5.  The  topical  arrangement  of  the  curriculum  has  been  per- 
manently abandoned,  and  either  a  non-topical  arithmetic  should 
accordingly  be  used,  or  some  good  modern  topical  arithmetic 
should  be  adopted  and  selections  should  be  carefully  made  so  as 
to  offer  a  moderate  "  spiral  "   ("  concentric-circle,"  "  recurring 
topic  ")  arrangement  adapted  to  the  growing  powers  of  the  child. 

52 


Certain  Great  Principles  of  Teaching  Arithmetic          53 

6.  No    extreme    of    "  method "    should  be    adopted    by    any 
teacher  or  school,  but  the  best  of  every  "  method  "  should  be 
known  as  far  as  possible  to  all.    To  measure  everything  in  sight, 
to  base  all  arithmetic  on  sticks,  to  go  to  extremes  on  number 
charts,  to  put  all  the  time  on  mental  arithmetic,  to  have  all 
written  work  placed  in  steps,  to  get  into  any  narrow  rut  what- 
ever,-— this  is  to  fail  of  the  best  teaching  and  to  narrow  the  hori- 
zon of  the  children  in  our  care.    A  good,  usable  text-book,  broad 
in  its  purpose,  modern  in  its  problems,  and  psychologically  ar- 
ranged, is  one  of  the  best  balance-wheels  on  us  all,  and  we  should 
depart  from  its  sequence  and  methods  only  for  reasons  that  have 
been  very  carefully  considered,   while  supplementing  its  good 
features  by  all  the  problems  with  local  color  that  we  can  find 
time  to  use. 

7.  Mental  (oral)  arithmetic  should  play  a  part  in  every  school 
year,   to  the  end  that  children  should  have  not  only  an  eye 
training  for  numerical  relations,  but  also  an  ear  training  and  a 
tongue  training.     The  text-book  may  be  expected  to  furnish  a 
considerable  amount  of  the  abstract  work  in  this  line,  but  the 
concrete  problems  may  well  be  correlated  with  local  life  and  with 
the  other  work  of  the  class. 

8.  Children's  analysis  instead  of  being  memorized  should  be 
genuine  statements  of  the  reasons  that  prompt  them  to  their 
solutions.    As  to  problems,  the  analysis  should  proceed  gradu- 
ally from  one-step  to  two-step  cases.    As  to  operations,  it  is  not 
to  be  expected  that  children  will  long  remember  the  reasons  in- 
volved;  they   should   understand  the   process   when  presented 
through  a  development  by  a  series  of  simple  questions ;  but  they 
should  not  be  expected  to  give  a  very  elaborate  explanation  of 
a  topic  like  long  division  after  the  process  is  once  understood. 

9.  Written  arithmetic  may  at  one  time  emphasize  the  rapid 
securing  of  results,  and  at  another  the  analysis  of  the  problem. 
Both  are  important,  but  in  general  the  accurate  result,  rapidly 
secured,  is  the  great  desideratum.    To  say  that  a  child  ought  not 
to  work  merely  for  the  answer  is  a  well-sounding  epigram,  but 
if  it  is  interpreted  to  mean  that  he  may  work  in  a  slovenly 
way,  dawdling  over  his  problem,  and  getting  an  answer  that  is 
absurd,  no  amount  of  neatly  written  step-work  can  atone  for  his 
mental  laziness. 


54  The  Teaching  of  Arithmetic 

10.  Arithmetic  should  be  as  interesting  as  any  other  subject 
in  school.    To  this  end  a  teacher  should  know  something  of  its 
interesting  story,  should  be  familiar  with  its  best  applications  to 
local  and  national  life,  should  know  how  to  treat  the  mental 
(oral)  exercises  in  sprightly  fashion,  and  should  have  a  fair  stock 
of  number  recreations. 

11.  The  improvements  in  the  technique  of  arithmetic,  includ- 
ing the  use  of  x  and  the  later  forms  of  operations,  should  be 
sympathetically  understood  by  teachers,  to  the  end  that  the  sub- 
ject may  not  stagnate  in  our  schools. 

12.  It  is  a  matter  of   relatively    little    importance    that    we 
present  fractions,  we  will  say,  in  this  way  or  that,  by  sticks  or 
paper- folding  or  clay  cubes  or  blocks;  but  it  is  a  matter  of  great 
importance  that  we  present  the  subject  in  some  concrete  fashion, 
to  the  end  that  the  child  does  not  proceed  by  arbitrary  rules  but 
makes  up  his  own  directions,  and  that  he  is  so  guided  that  these 
are  the  best  that  can  be  evolved  at  his  age.     What  has  been 
rather  pedantically  called  "  heuristic  teaching,"   in  its.  original 
form  as  old  as  Socrates  at  least,  should  always  be  in  the  teacher's 
mind — to  lead  the  child  unconsciously  to  feel  that  he  is  the  dis- 
coverer, but  to  see  to  it  that  he  is  allowed  to  discover  and  to 
fix  in  mind  only  what  the  world  has  found  to  be  the  best.    The 
carrying  out  of  this  policy,  in  arithmetic  or  any  other  subject, 
is  one  of  the  essentials  of  good  teaching. 

These  are  a  few  of  the  larger  principles  that  should  guide  the 
teacher  of  arithmetic.  The  list  might  be  extended,  but  these 
suffice  to  show  the  spirit  in  which  the  subject  should  be  ap- 
proached. Minor  principles  will  appear  as  we  consider  the  work 
of  the  various  grades  or  school  year. 

BIBLIOGRAPHY:  Smith,  Teaching  of  Elementary  Mathematics; 
Young;  The  Teaching  of  Mathematics;  De  Morgan,  On  the 
Study  and  Difficulties  of  Mathematics,  Chicago,  1898;  Clifford, 
Common  Sense  of  the  Exact  Sciences,  3d  edition,  New  York, 
1892. 


CHAPTER  XIII 
GENERAL  SUBJECTS  FOR  EXPERIMENT 

It  is  well,  before  leaving  this  general  discussion,  to  consider 
a  few  of  the  subjects  for  legitimate  experiment  in  the  teaching 
of  arithmetic,  that  might  occupy  the  attention  of  schools  of 
observation  or  practice  in  connection  with  institutions  for  the 
training  of  teachers. 

(i)  It  is  desirable  to  know  just  how  far  recreations  in  num- 
ber can  be  used  to  advantage  in  teaching  arithmetic.  Of  course 
we  have  such  games  as  bean  bag,  ring  toss,  and  sometimes  dom- 
inoes and  number  games  with  cards,  used  in  the  school  room. 
This,  however,  is  a  mere  beginning.  There  are  many  more 
games  that  are  usable  for  children.  For  example,  more  people 
in  the  history  of  this  world  have  learned  elementary  number 
through  dice  than  in  the  public  school,  and  this  is  only  one  of 
several  widely  used  number  games.  It  would  be  very  easy  to  go 
to  a  ridiculous  and  even  dangerous  extreme  in  this  matter,  and 
a  teacher  who  begins  to  work  upon  it  will  naturally  tend  to 
do  this,  and  will  need  a  balance  wheel  upon  his  endeavors. 
Nevertheless  the  work  has  never  yet  been  done  scientifically 
and  it  ought  to  be  undertaken.  This  is,  however,  only  a  small 
part  of  the  problem.  There  is  the  whole  field  of  mathematical 
recreations  that  must  sometime  be  examined  scientifically.  We 
have  a  large  but  undigested  literature  upon  the  subject,  and  no 
one  has  ever  yet  studied  it  from  the  standpoint  of  the  definite 
needs  of  the  grades.  The  result  of  such  a  study  ought  to  add 
greatly  to  the  interest  in  arithmetic,  without  going  to  any  ridicu- 
lous and  impractical  extreme.  Work  there  must  always  be  in 
arithmetic,  and  it  ought  to  be  good  hard  work,  but  there  is  no 
reason  why  we  should  not  let  pupils  see  the  amusements  as  well 
as  the  other  interesting  phases  of  the  subject. 

In  this  matter  of  the  play  element  the  following  brief  list  of 
games  available  for  primary  number  work  may  be  of  service. 

55 


56  The  Teaching  of  Arithmetic 

It  was  prepared  by  Miss  Julia  Martin  in  connection  with  some 
work  undertaken  for  the  State  Department  of  Public  Instruction 
of  Michigan: 

(1)  The  Game  of  Tag.     Every  child  in  the  room  is  given  a  number. 
One  child  is  the  leader.    He  gives  orally  any  number  below  a  number  that 
has  been  designated  by  the  teacher.    All  pupils  who  have  numbers  which 
are  factors  of  the  given  number  must  change  seats.    They  may  be  tagged 
while  running  or  if  they  fail  to  run. 

(2)  The  Game  of  "Simon  says  Thumbs  Up."    The  teacher  or  one  of 
the  pupils  may  act  as  leader.    The  whole  class  take  a  position  with  thumbs 
up,  and  each  pupil  has  a  number.    The  leader  says  "  Simon  says  15."    The 
thumbs  of  numbers  3  and  5   (the  factors)  must  go  down.     "  Simon  says 
12."    The  thumbs  of  2,  3,  4,  and  6  must  go  down.     Occasionally  the  leader 
says  "  Simon  says  all  thumbs  up."    This  game  may  be  used  in  addition 
and  subtraction  drill  work. 

(3)  The  Guessing  Game.    This  game  is  suitable   for   Grade   II.     The 
materials  needed  are  20  counters  for  each  pupil  and  the  teacher,  and  a 
slip  of  paper  and  a  pencil  for  each.    The  teacher  picks  up  a  handful  of 
counters  and  says,  "  How  many  counters  have  I  ?  "    The  pupil  guesses,  and 
if  correct  he  gets  the  counters,  and  writes  the  number  on  a  slip  of  paper. 
If  he  fails  to  give  the  correct  number  he  must  give  the  teacher  the  differ- 
ence between  the  number  and  his  guess.    He  must  write  this  number  on 
the  paper.    At  the  close  of  the  game  the  pupils  add  the  gains  and  losses, 
and  check  their  work  by  the  counters  left. 

Miss  Martin  suggests  the  following  additional  list  as  valuable 
in  arithmetic :  buzz ;  fizz ;  railroad  station ;  bean  bag ;  days  of  the 
week;  hop  Scotch;  school  ball;  bird  catcher;  marbles.  To  this 
list  may  also  be  added  certain  card  games  of  an  arithmetical 
nature  published  under  the  editorship  of  the  author  by  the 
Cincinnati  Game  Company. 

(2)  It  is  desirable  definitely  to  map  out  the  chief  interests  of 
children  from  grade  to  grade,  with  a  view  to  ascertaining  the  best 
field  for  applied  problems  from  year  to  year.  We  know  these 
interests  in  a  general  way,  and  we  know  the  child's  mind  well 
enough  to  judge  of  his  arithmetical  powers  from  year  to  year. 
But  we  do  not  yet  know  these  interests  in  the  exact  way  that  we 
should  know  them.  For  example,  when  is  the  game  element 
strongest?  When  does  the  interest  in  the  heroic  become  most 
pronounced,  and  is  the  period  the  same  for  boys  as  for  girls  so 
that  we  may  use  this  information  in  problem  work  in  a  mixed 
school?  When  does  the  interest  become  manifest  in  the  food 
supply  of  our  country?  When  in  the  clothing  supply?  When 


General  Subjects  for  Experiment  57 

in  transportation?  When  in  the  mines?  When  in  manufactur- 
ing? When  in  commercial  life?  We  know  all  this  in  a  general 
way,  but  only  so, — not  exactly,  not  as  the  result  of  any  scientific 
investigation.  And  when  we  do  know  this  the  rational  appli- 
cations of  arithmetic  from  year  to  year  will  be  so  much  better 
understood  that  the  subject  will  have  an  interest  that  is  now 
but  feebly  developed,  and  a  value  that  we  at  present  appreciate 
only  in  part. 

(3)  We  also  need  to  know,  statistically  if  possible,  the  result 
upon  a  group  of  children  of  emphasizing  the  abstract  problem; 
upon  another  group  of  emphasizing  the  concrete;  and  upon  a 
third  of  leaving  the  two  in  about  the  balance  that  experience  has 
dictated.    We  have  had  some  scientific  investigation  in  this  line, 
but  it  has  been  very  slight  and  it  is  therefore  not  conclusive. 
To  emphasize  the  concrete  would  be  to  diminish  the  number  of 
problems  very  greatly,  and  it  would  seem  to  give  less  exercise  in 
number  relations,  and  it  does  seem  to  give  less  satisfactory  re- 
sults.    On  the  other  hand  it  may  create  interest  in  the  work, 
thereby  increasing  the  power  of  impression  of  the  number  rela- 
tions, so  that  because  the  child  multiplies  only  V25  as  many  times 
it  will  not  follow  that  he  knows  his  subject  only  1/23  as  well. 
At  present  the  whole  subject  is  in  the  domain  of  doctrinaire 
argument;  what  is  needed  is  a  scientific  investigation  of  the 
problem  by  some  school  or  some  person  who  is  unbiased  in 
the  case. 

(4)  We  are  at  present  entirely  unsettled  upon  the  question  of 
time  to  be  assigned  to  arithmetic.    Two  scientific  investigations 
have  been  made,  but  each  is  incomplete.     It  would  seem  that 
excellence  in  arithmetic  work  is  much  less  a  function  of  the 
time  assigned  to  it  than  has  formerly  been  supposed.     Such  an 
investigation  would  probably  require  a  number  of  years  for  its 
satisfactory    completion,  but   it   might   be   undertaken    in    any 
particular  school  system  in  a  single  year  with  some  very  helpful 
results. 

(5)  We  are  quite  uncertain  as  to  the  relative  amount  of  time 
to  be  devoted  to  oral  and  written  arithmetic  in  our  schools. 
The  wave  of  oral  work,  beginning  with  Pestalozzi  and  culminat- 
ing in  this  country  with  Warren   Colburn   as    mentioned    in 
Chapter  VII,  gradually  subsided  some  years  ago.    What  should 


58  The  Teaching  of  Arithmetic 

we  be  doing  in  the  matter  to-day?  It  is  very  easy  to  talk  dog- 
matically about  it,  but  we  need,  if  it  is  possible,  a  scientific 
investigation  as  to  the  practical  results  of  more  "  mental "  arith- 
metic, and  of  less.  Would  it  be  well  to  have  the  work  much 
more  oral  than  at  present?  or  would  we  gain  by  confining  our 
energies  more  to  written  work?  Is  there  any  scale  by  which 
we  can  definitely  measure  this  matter?  and  if  so,  what  is  it  and 
what  are  the  results  of  the  measurement? 

(6)  Just  how  far  we  are  justified  in  departing  from  the  old 
plan  of  making  the  operations  the  basis  for  a  course  in  arith- 
metic, and  of  substituting  therefor  the  applications?  Shall  we 
ever  be  justified  in  giving  up  multiplication  as  a  topic  and  in 
substituting  a  chapter  on  housebuilding  that  shall  bring  in 
multiplication  as  needed?  Of  course  in  the  latter  part  of  arith- 
metic we  put  the  application  to  the  front,  as  in  the  early  part 
we  put  the  operation  there;  but  just  where  we  should  draw  the 
line,  and  how  far  should  this  latter  plan  encroach  upon  the 
former? 

To  this  list  of  general  subjects  for  experiment  my  colleague 
Professor  Henry  Suzzallo  contributes  a  number  of  others,  and 
these  make  up  the  rest  of  this  chapter.  He  remarks  in  the 
first  place  that  it  is  not  sufficient  that  a  new  way  or  an  old  way 
of  teaching  has  succeeded,  e.  g.,  in  the  addition  of  fractions. 
The  test  of  the  worth  of  a  given  method  is  not  alone  that  it 
gets  a  thing  done  efficiently ;  it  must  get  it  done  as  economically 
as  possible.  The  method  of  most  worth  is  the  one  that  obtains 
the  efficient  result  with  the  least  possible  expenditure  of  energy. 
The  comparative  worth  of  two  methods  must  be  investigated 
under  experimental  conditions,  with  children  of  about  the  same 
grade,  age,  and  previous  training,  taught  by  teachers  of  fairly 
even  strength  of  personality,  so  that  approximately  the  only 
difference  in  the  conditions  of  the  trial  is  the  difference  in  the 
methods  involved  in  the  test.  Finally  the  repetition  of  the  ex- 
periment in  more  than  one  school  or  system,  reduces  the  danger 
always  present  in  assuming  that  a  special  group  of  children  and 
teachers  is  typical  of  school  conditions  in  general.  A  sample 
experiment  will  make  the  method  of  investigation  clear,  as 
follows : 

In  the  teaching  of  addition  combinations  it  is  the  requirement 


General  Subjects  for  Experiment  59 

of  certain  courses  of  study  that  combinations  and  their  reverses 
be  taught  in  association  with  each  other.  The  teaching  of  "  3 
and  2  are  5,"  should  be  followed  at  once  by  the  learning  of 
"  2  and  3  are  5."  In  other  courses  of  study  or  texts  there 
naay  be  quite  an  interval  between  the  learning  of  these  two  com- 
binations. In  fact  the  student  may  learn  each  separately  with- 
out being  conscious  of  any  greater  intimacy  between  these  two 
combinations  than  between  any  other  two,  e.  g.,  "  6  and  3  are 
9  "  and  "  4  and  2  are  6."  Those  who  advocate  the  first  method, 
imply,  if  they  do  not  expressly  state,  that  learning  a  combina- 
tion and  its  reverse  simultaneously  is  more  efficient  than  learning 
them  separately  and  unrelated.  The  problem  for  the  investigator 
is  to  determine  whether  or  not  this  is  true. 

For  example,  in  any  large  city  school  there  may  be  two, 
three,  four,  or  more  classes  of  one  grade  in  which  combinations 
in  addition  are  taught  for  the  first  time.  In  the  case  where  there 
are  four  classes,  two  could  be  set  off  against  the  remaining  two, 
with  such  judgment  as  to  equalize  number  and  quality  of  grades, 
teachers'  personalities,  and  other  factors,  in  so  far  as  they  may 
be  equalized  within  such  a  limited  range.  All  four  classes  could 
then  be  given  the  same  list  of  combinations  to  be  learned,  and 
a  common  method  of  general  procedure  could  be  laid  down, 
by  the  experimenting  principal,  the  only  general  difference  in 
procedure  being  that  one  group  of  classes  would  always  learn 
the  reverses  immediately  following  the  original  combination  to 
which  it  is  related,  while  the  other  group  would  learn  them  in 
an  order  that  would  separate  the  original  combination  and  its 
reverse.  Thus : 

Group  I. 

223243253435445 
+232423524353454 


455666777788899 
Group  II. 

222323345345455 
-r2  34354544222334 

456677889567789 


6o  The  Teaching  of  Arithmetic 

An  equal  amount  of  time  having  been  spent  in  all  the  classes 
up  to  a  point  where  they  have  approximately  covered  the  com- 
plete list  of  combinations,  a  test  could  be  given  to  determine 
how  far  each  individual  had  mastered  these  number  facts.  After 
a  lapse  of  a  week  or  ten  days  another  examination  would  show 
how  stable  the  mastery  had  been  in  each  case.  Any  difference 
between  the  group  taught  by  one  method  and  the  group  taught 
by  the  other  method  as  clearly  shown  by  statistical  interpreta- 
tion would  then  tend  to  indicate  the  relative  efficiency  of  the 
two  methods. 

This  being  the  method  of  experimentation,  a  few  of  the  gen- 
eral questions  may  now  be  considered.  First,  does  the  effective 
Use  of  objective  work  demand  a  large  amount  of  this  work 
with  relatively  young  students  and  a  smaller  amount  with  rela- 
tively old  students?  Or  should  the  distribution  of  objective 
work  be  determined  by  the  fact  of  ignorance  or  immaturity  in 
any  special  phase  of  arithmetic  regardless  of  the  age  of  the  stu- 
dent ?  To  what  extent  does  general  arithmetical  maturity  require 
less  objective  work  in  the  first  formal  study  of  fractions  than 
in  the  first  formal  study  of  addition? 

Do  all  of  the  fundamental  combinations  in  any  given  field 
(addition,  multiplication,  etc.)  require  objective  development? 
How  many  combinations  need  to  be  developed  objectively 
before  the  child  will  clearly  know  that  succeeding  combinations 
given  by  the  teacher  authoritatively  stand  for  real  relations? 

Furthermore,  to  what  extent  may  the  constant  handling  of 
objects,  pictures,  and  diagrams,  and  concrete  imaging  in  gen- 
eral, interfere  with  rapid  abstract  manipulation  of  numbers  and 
number  combinations? 

To  what  degree  are  the  difficulties  of  children  with  arithmetic 
problems  due  to  a  failure  to  understand  underlying  concrete 
situations,  becaus*e  they  do  not  understand  the  language  by 
which  it  is  intended  to  convey  them?  What  is  the  relative  diffi- 
culty in  understanding  the  significance  of  a  situation  when  the 
presentation  is  (i)  objective,  (2)  oral,  and  (3)  written?  Would 
it  be  well  to  postpone  the  written  presentation  of  problems  until 
a  specific  number  of  school  years  of  language  training  have 
been  given  ? 

How  far  is  it  necessary  to  develop  a  special  terminology  for 


General  Subjects  for  Experiment  61 

school  use  in  the  subject  of  arithmetic,  the  terms  being  little 
used  in  ordinary  social  relations?  For  example,  consider  the 
case  of  the  words  multiplicand  and  dividend,  the  latter  having 
a  radically  different  meaning  in  business  life.  In  the  case  of 
signs  consider  such  semi-algebraic  symbols  as  -T-,  and  even  +. 
If  special  signs  are  used  in  examples,  to  stand  for  the  process 
of  calculation  demanded  by  the  situation  (which  might  have 
been  expressed  in  concrete  problem  form)  how  wide  and  varied 
should  the  language  of  problems  describing  situations  be?  If 
+  may  be  used  in  an  example,  while  the  same  relation  is  ex- 
pressed in  a  problem  by  such  words  and  phrases  as  "  added," 
"  and,"  "  together,"  "  how  many,"  "  altogether,"  how  wide 
should  the  latter  vocabulary  be?  To  what  extent  is  a  lack  of 
dealing  with  problems  presented  through  varied  language,  ex- 
planatory of  the  failure  of  children  in  problem  tests  given  by  an 
outsider,  another  teacher,  the  principal  or  the  superintendent, 
when  the  children  had  always  "  done  perfectly  "  the  problems 
that  their  own  teacher  gave  them? 

Should  long  forms  of  expressed  calculations  always  precede 
short  forms?  Taking  the  specific  cases  of  division  by  a  one- 
figure  divisor,  or  column  addition  involving  several  places,  does 
the  fact  that  the  long  form  is  the  one  really  used  with  a  two- 
figure  divisor  make  the  case  different  from  addition  of  several 
two-figure  addends,  where  the  long  form  is  really  not  used  in 
business  ? 

To  what  extent  are  so  called  "  aids  "  or  "  crutches  "  real  helps 
or  final  hindrances  to  efficient  and  rapid  work,  as  in  the 
case  where  numbers  are  crossed  out  and  others  written  in  their 
stead  throughout  the  process  of  "  borrowing "  in  column  sub- 
traction ? 

Which  types  of  problems  are  of  most  concrete  interest  to  the 
child,  ( i )  those  drawn  from  his  own  spontaneous  play  and  work 
life,  or  (2)  those  drawn  from  the  facts  of  actual  social  life  about 
him?  Are  both  essential  in  achieving  the  aims  of  arithmetic 
teaching?  If  so  is  there  any  special  law  of  advantageous  usage 
of  each  of  the  two  types?  Should  problems  from  his  own  life 
be  used  to  introduce  a  field  showing  the  necessity  for  learning 
means  of  calculation,  and  those  from  social  life  be  used  for  fur- 
ther, later,  and  final  application  of  the  formal  processes  of  cal- 


62  The  Teaching  of  Arithmetic 

dilation  that  have  been  mastered?  Does  it  make  a  problem 
concrete  to  the  child  merely  because  it  is  a  concrete  reality  exist- 
ing in  the  world?  May  not  an  imagined  problem  vividly  within 
the  grasp  of  his  own  imagination,  be  more  concrete  in  interesting 
the  pupil  in  its  solution  than  one  which  actually  exists  in  the 
real  world  ? 

Problems  may  be  done  by  the  pupil  (i)  silently  ("mental 
arithmetic"),  (2)  orally,  (3)  in  written  form  on  paper  or  black- 
board, and  (4)  with  a  mixture  of  any  two  or  all  three  of  the 
preceding  methods.  Which  methods  represent  final  forms  in 
which  efficiency  is  demanded  in  ordinary  life?  Which  forms 
merely  represent  transitional  means  used  by  the  teacher  to  keep 
track  of  the  workings  of  the  child's  mind?  What  is  the  proper 
order  and  emphasis  of  these  forms  in  the  mastery  of  a  single 
new  line  of  work,  say  in  a  problem  where  the  child  needs  first 
to  divide  and  then  to  multiply? 

To  what  extent  do  precise  oral  forms  assist  in  the  correct 
analyses  of  problems,  e.  g.,  "  If  two  apples  cost  6  cents,"  etc.  ? 
To  what  extent  do  precise  oral  forms  assist  in  the  memorization 
of  combinations  or  manipulations,  e.  g.,  "  3  4's  are  12,"  "  put 
down  the  three  and  '  carry  '  the  2  "  ?  On  the  other  hand,  to  what 
extent  do  precise  written  arrangements  of  analyses  assist  the 
child  in  carrying  out  a  strictly  logical  mode  of  thinking?  Take 
the  following  case  as  an  example:  "3  pencils  cost  I5c. 
i  pencil  costs  1/3  of  I5c."  etc. 

Are  certain  algorisms  more  efficient  and  economical  than 
others?  In  which  form  should  a  pupil  learn  his  addition  com- 

6 
binations,  4  -f  6  =  10   or  +  4?    In  the  case  of  long  division  should 

_IO 
it  be  45)6;836(/- or  45)67836? 

Is  it  wise  to  use  several  different  algorisms  for  a  single 
process,  when  it  is  possible  to  use  but  one?  In  other  words 
does  not  a  multiplication  of  algorisms  increase  the  amount  of 
memorization  of  forms  required  of  the  child?  In  division  instead 
of  having  three  algorisms,  one  for  the  combinations  6-^-3  =  2, 
another  for  short  division  3)45735,  and  a  third  for  long  division 
345)45735,  would  it  be  any  better  to  have  a  similar  form  through- 


General  Subjects  for  Experiment  63 

out  for  easy  identification  of  the  three  processes  as  fundamentally 
one,  thus 

_2  15245  132 

3)6,  3)45735,  345)45735 

345 

1123 
1035 


885 
690 

195 

and  in  connection  with  all  this  what  weight  should  be  given  to 
Professor  Smith's  objection  that  the  universal  custom  of  the 
business  world  does  not  recognize  the  first  two  forms  of  this 
latter  set? 

Shall  the  attempt  to  get  rapidity  of  calculation  be  preceded 
by  the  attempt  to  get  absolute  accuracy,  the  quickening  of  the 
work  being  left  until  certainty  of  command  over  combinations 
is  assured?  Or  shall  rapidity  in  handling  combinations  and 
manipulations  be  a  parallel  activity?  Take  for  example  the  at- 
tempt to  have  the  children  attack  the  successive  combinations 
in  column  addition  with  a  definite  rhythm,  the  teacher  pointing 
to  each  successive  stage,  or  chorus  work  being  utilized  with  the 
quickest  students  setting  the  pace? 

Does  motor  activity  accompanying  a  process  of  memorization 
of  combinations  require  fewer  repetitions  than  where  no  special 
provision  is  made  for  motor  activity?  How  far  does  it  help 
to  repeat  the  numbers  aloud  ?  to  write  them  on  paper  ?  to  manipu- 
late objects  when  the  combination  is  being  learned? 

How  far  can  rhythm  be  used  in  memorizing  combinations  or 
tables?  Does  the  use  of  rhythm  decrease  the  number  of  repeti- 
tions required  for  mastery?  How  much? 

BIBLIOGRAPHY:  On  the  general  question  consult  the  works  of 
Professor  Young  and  the  author.  On  the  special  question  of 
number  games  consult  the  following: 

Johnson,  G.  E.,  Education  by  Plays  and  Games.  Pedagogical  Semi- 
nary (Oct.  1894),  Vol.  III.  This  work  contains  a  list  of  five  hundred 
games  for  children. 


64  The  Teaching  of  Arithmetic 

Johnson,  G.  E.,  Education  by  Plays  and  Games,  Ginn  and  Company. 
This  book  contains  a  suggestive  course  of  plays  and  games,  and  these 
games  are  correlated  with  the  regular  subjects  of  the  curriculum. 

Harper,  C.  A.,  One  Hundred  and  Fifty  Gymnastic  Games,  G.  H.  Ellis, 
Boston.  This  and  the  two  preceding  are  the  most  important. 

Grey,  Marion,  Two  Hundred  Indoor  and  Outdoor  Games,  Freidenker 
Publishing  Company,  Milwaukee. 

Newell,  W.  W.,  Games  and  Songs  of  American  Children.  Harper  and 
Brothers,  New  York. 

Kingsland,  Mrs.  Burton.  The  Book  of  Indoor  and  Outdoor  Games, 
Doubleday,  Page  &  Co. 


CHAPTER  XIV 
DETAILS  FOR  EXPERIMENT 

Besides  these  subjects  that  may  be  designated  as  more  or  less 
general,  there  are  many  details  that  demand  investigation.  These 
are  partly  arithmetical  and  partly  psychological  in  nature,  and 
belong  quite  as  much  in  one  field  as  another.  So  far  as  the 
investigation  itself  is  concerned,  the  trained  psychologist  is  the 
only  one  who  could  be  expected  to  secure  satisfactory  results. 
Some  of  these  details  have  already  been  mentioned  in  this  article, 
and  a  dogmatic  opinion  has  been  expressed  concerning  them. 
It  is  proper,  however,  to  set  them  forth  more  at  length  for 
the  use  of  investigators. 

At  my  request  Professor  Suzzallo,  who  has  given  the  matter 
much  attention,  has  supplemented  the  suggestions  given  in  the 
preceding  chapter  by  a  further  list  of  such  special  experiments 
as  occur  to  him,  and  has  kindly  permitted  me  to  embody  them 
in  this  article.  I  therefore  insert  them  without  comment,  it  being 
desirable  that,  in  this  place  at  least,  they  should  appear  with  as 
little  expression  of  opinion  as  possible.  This  chapter  XIV  is, 
therefore,  to  be  considered  as  Professor  Suzzallo's. 

The  quarrels  that  exist  as  to  the  proper  methods  of  teaching 
arithmetic  are  not  merely  theoretic.  Every  difference  in  practice 
in  the  treatment  of  a  subject  in  arithmetic  implies  a  difference  of 
opinion,  and  consequently  a  controversy.  It  is  in  the  settlement 
of  these  practical  controversies  that  careful  experimental  methods 
can  be  of  large  service.  It  can  scarcely  be  said  that  such  an  ap- 
proximately scientific  approach  has  been  seriously  attempted 
even  by  educational  theorists,  and  much  less  by  school  principals 
and  superintendents.  That  there  are  obvious  reasons  for  this 
fact  goes  without  saying.  The  average  teacher  or  principal  is 
too  busy  with  other  matters,  which  for  the  time  being  are 
exceedingly  urgent.  But  it  must  be  equally  obvious  that  pro- 
vision for  the  investigation  of  teaching  problems  must  be  made 

65 


66  The  Teaching  of  Arithmetic 

somewhere  within  the  profession.  Perhaps  in  the  beginning,  it 
must  be  left  to  those  scattered  workers  who  have  at  once  the 
impulse  and  the  opportunity  to  conduct  investigations.  In  the 
hope  of  being  of  assistance  to  such  as  these,  the  following  list  of 
practical  problems  now  existing  in  the  teaching  of  primary  arith- 
metic has  been  prepared.  The  list  is  confined  to  the  handling 
of  whole  numbers  in  the  first  few  grades,  and  is  by  no  means 
complete.  It  pretends  merely  to  suggest  certain  differences 
of  practice,  the  relative  value  of  which  needs  to  be  determined 
by  something  more  than  mere  opinion. 

(1)  Which  is  the  best  way  to  teach  young  children  to  count 
serially  from  I  to  100?    To  have  them  count  by  ones  from  the 
beginning,  extending  the  series  as  fast  as  the  child  can  memorize 
the  same,  without  any  conscious  effort  in  the  direction  of  show- 
ing the  child  that  the  series  repeats  with  a  certain  regularity 
after  twenty  is  passed?    Or  to  have  them  memorize  the  names 
in  their  order  from  one  to  thirty  (by  which  time  the  regularity 
is  established  as  a  basis)   and  then  have  them  learn  to  count 
by  tens,  later  using  the  counting  by  ones  and  the  counting  by 
tens  as  a  double  basis  for  learning  to  count  serially  from  thirty 
to  one  hundred? 

(2)  Assuming  that  oral  counting  leads  mainly  to  the  associa- 
tion of  a  name  (27)  with  a  given  position  in  a  series  of  names 
(between  26  and  28),  how  far  is  it  advisable  for  a  number  to  be 
associated  with  a  given  idea  of  mass  or  grouping,  as  when  the 
device  of  two  bundles  of  ten  sticks  each  and  seven  individual 
sticks  is  used  to  explain  27?    Does  the  effort  toward  the  asso- 
ciation of  concrete  images  and  numbers  ultimately  interfere  with 
the  rapid  manipulation  of  figures  in  complex  calculations  ?    How 
far  does  the  material  in  objective  work  need  to  be  varied  with 
first-grade  children  (sticks,  lentils,  boys,  etc.)  so  that  the  idea 
associated   with  a  number   shall   be   abstract   rather  than  the 
image    of    any    particular    concrete    thing    or    group    of    con- 
crete things? 

(3)  How  far  is  group  counting  (counting  by  2's,  3*5,  etc.) 
really  counting,  that  is,  proceeding  from  one  number  to  another 
by  an  act  of  absolute  memory  (saying  3,  6,  9,  etc.,  exactly  as 
one.  says  i,  2,  3,  etc.)  ?    How  far  is  it  really  a  process  of  con- 
secutive adding  (3  and  3  are  6,  6  and  3  are  9,  etc.)  ?    If  it  is  a 
mixture  of  both,  where  does  one  process  end  and  the  other 


Details  for  Experiment  67 

begin?  If  group  counting  is  really  adding,  should  it  not  always 
be  classified  with  the  work  of  addition,  and  placed  so  as  to  assist 
it,  rather  than  be  operated  independently  as  alleged  counting? 
How  far  is  group  counting  as  real  counting  desirable?  How  far 
may  it  be  used  as  another  form  of  addition?  In  the  latter  case 
should  it  precede  or  follow  combination  work  in  addition  (6,  9, 
12,  etc.,  precede  or  follow  6  +  3  =  9,  9  +  3=i2>  etc.)?  If 
counting  forward  is  an  aid  to  addition,  how  far  can  counting 
backward  be  an  aid  to  subtraction?  How  far  is  real  counting 
backward  (by  sheer  act  of  consecutive  memory)  of  valid 
social  use? 

(4)  How  far  shall  the  three  processes  of  (i)  oral  counting, 
(2)  reading  of  numbers,  and  (3)  writing  of  numbers,  be  parallel 
in  the  first  year  of  formal  arithmetic  teaching?  Should  counting 
precede  reading,  and  reading  precede  writing  of  numbers  ?    How 
far  are  they  dependent  upon  each  other?    In  relation  to  accom- 
plishment in  any  one  of  these,  when  should  the  teaching  of  the 
other  begin? 

(5)  Why  do  young  children  who  know  their  numbers  up  to 
twenty  write  16  correctly  at  first,  and  later,  when  they  are  sup- 
posed to  know  their  numbers  to  100,  write  16  as  61  ?    Would 
further  and  special  drill  on  certain  numbers  of  the  series  have 
prevented  this  error?    Which  numbers  require  this  special  care? 
Why  do  children  sometimes  say  "  five-teen  ?  " 

(6)  In  teaching  children  to  read  and  write  numbers,   how 
far  is  it  useful  and  how  far  is  it  confusing  to  have  them  know 
the  place  names  (unit  of  units,  tens  of  units,  hundreds  of  units, 
etc.)  ?    Should  such  a  classification  be  given  to  the  child  finally, 
or  not  at  all  ?     Is  the  so-called  method  of  "  group  reading " 
superior  to  the  "  place  "  method  ?      To  the  method  of  direct 
memorization  ?    In  the  "  group  "  method  a  child  reads  and  writes 
all  his  numbers  as  he  would  numbers  of  three  figures  or  less, 
naming  them  from  the  commas  which  mark  off  the  groups  of 
three,  as  in  34,026,  "  34  "  =  "  thirty-four,"  "  ,  "  =  "  thousand," 
"  026  "  =  "  twenty-six."    What  are  the  special  errors  which  are 
peculiar  to  the  "  place  "  method  ?    What  are  the  special  errors 
peculiar  to  the  "  group  "  method  ? 

(7)  Are  all  numbers  of  from  four  to  six  places  equally  easy 
to  read  and  write?     If  not,  what  are  the  types  representing 
gradations  of  difficulty?    Taking  the  following  types: 


68  The  Teaching  of  Arithmetic 

4,000           In  which  are  errors  most  frequent?    When  these 
80,000      same  figures  appear,  not  in  "  thousands  place  "  but 
13,000       in  "  units  place,"  would    the    order    of    difficulty 
257,000      be  the  same  or  different?    As  in  1,000 
900,000  i,257 

120,000  i,9°° 

304,000  i,  1 20 

1,304 
1,013 
1,004 

i, 080?      It   will 

be  noted  that  there  are  seven  types  in  the  first  list  and  eight  in 
the  second,  due  to  the  introduction  of  ,000.  Note  also  that  4, 
becomes  ,004  in  the  second  list,  and  passes  from  the  easiest  to 
next  to  the  most  difficult.  Are  such  distinctions  characteristic 
of  children's  experiences  with  numbers? 

How  does  some  provision  for  equalizing  drill  in  all  types  of 
numbers  minimize  the  unequal  distribution  of  errors,  as  opposed 
to  the  hit-and-miss  methods  of  drilling  from  personal  lists  made 
up  by  the  teacher  as  he  needs  them? 

According  to  the  types  enumerated  above  as  a  result  of  the 
investigation  of  thousands  of  children's  papers,  would  there 
not  be  56  (7X8  =  56)  drill  types  for  thousands,  and  448 
(7X8X8  =  448)  for  numbers  in  millions  place? 

(8)  It  is  generally  said  that  there  are  forty-five  fundamental 
combinations  which  are  the  basis  of  all  work  in  addition.    What 
are  the  fundamental  facts  that  are  required  as  basic  and  which, 
once  learned,  may  be  applied  in  new  forms  and  situations  over 
and  over  again? 

There  are  ten  numbers,  from  o  up  to  9.  Each  of  these  may  be 
combined  with  itself  and  the  nine  others,  thus  making  100  combi- 
nations, from  0  +  0  =  0  up  to  9  +  9  =  18.  The  19  zero  com- 
binations are  left  out,  leaving  81  combinations.  Of  the  81 
remaining,  36  are  reverses  (2  +  7  =  9  '1S  a  reverse  of  7  +  2  =  9). 
Omitting  these  there  are  45  combinations  left  as  fundamental. 
Is  this  procedure  correct? 

(9)  How  far  does  the  learning  of  7  +  2  =  9  also  guarantee 
the  acquiring  of  its  reverse,  2  +  7  =  9?     Will  the  second  be 
known  without  further  drill?    With  how  many  less  repetitions 


Details  for  Experiment  69 

will  it  be  learned  because  the  other  combination  is  mastered? 
Will  the  two  combinations  mentioned  be  learned  with  fewer  repe- 
titions when  they  are  constantly  learned  together,  as  opposed  to 
being  learned  as  separate  individual  combinations  the  relation  of 
which  is  not  specially  kept  in  mind? 

(10)  Is  there  a  justification  for  saying  that  the  zero  combina- 
tions (0  +  3  =  3)  mav  be  omitted  as  not  being  basic?  The 
contention  is  that  they  never  occur  in  single  combination.  No 
one  says,  "  I  have  nothing  and  three,  and  adding  them  I  have 
three."  In  such  a  situation  we  merely  count  what  we  have,  we 
do  not  add  our  count  to  what  we  do  not  have,  for  we  are  not 
conscious  of  the  latter  numerically. 

But  may  not  the  zero  combinations  be  necessary  for  their  later 
application  in  column  addition ?  6  +  3  =  9  is  used  as  16  +  3=  19 
and  0  +  4  =  4  is  used  as  10  +  4  =  14.  Is  it  true  that  all  zeros 
in  column  addition  are  ignored? 

In   four  conceivable  cases,    o  +  4  =    4 

4  +  0=    4 
10  +  4=  14 

14  +  o  =  14    where   the    zero     is 

found  in  column  addition,  it  may  be  said  that  in  three  the  zero  is 
treated  with  one  attitude ;  it  is  ignored.  In  the  case  of  10  +  4=14 
the  zero  is  treated  as  part  of  quantity,  and  must  be  learned.  Must 
not  the  child  know  all  the  applied  zero  combinations  from 
10+2=12  up  to  10  +  9=19,  and  must  not  these  eight  combi- 
nations be  provided  somewhere  in  the  child's  instruction  ?  Which 
then  is  the  most  economical  and  efficient  way  of  teaching  the  zero 
combinations  mentioned  ?  To  teach  them  as  0  +  4  =  4  and  then 
apply  as  10  +  4=  14,  or  to  teach  as  10  +  4=  14  from  the  very 
beginning?  Experimentation  ought  to  reveal  the  relative  value 
of  the  two  methods.  It  ought  to  reveal  the  difference  between 
making  some  provision  for  them  and  making  no  formal  provision. 

(n)  In  the  list  of  forty-five  fundamental  combinations,  the 
zero  combinations  were  left  out  (when  some  should  probably  have 
been  left  in)  and  the  combinations  with  one  (6+1=7)  were 
left  in.  Should  they  also  have  been  left  in?  As  no  one  adds  o 
to  a  number  in  a  single  combination  in  actual  life,  it  might  be 
asked  if  we  ever  add  one?  We  really  count  one  more,  not  add. 
When  we  have  6  and  i  more,  do  we  not  count  6,  7,  nor  add  6  +  I 


70  The  Teaching  of  Arithmetic 

=  7.  Counting  is  a  more  fundamental  habit  than  adding,  and  it 
is  contended  that  when  i  is  met  in  any  column,  the  mind  really 
climbs  the  scale  i,  it  does  not  group  it  as  where  3  is  met.  If 
this  is  so  the  children  being  able  to  count  serially  already,  need 
not  learn  one  as  an  addition.  This  would  omit  17  combinations. 
Experimentation  would  show  how  far  children  taught  the  com- 
binations with  i  were  superior  in  column  addition  where  I's 
occurred,  to  children  who  had  not  had  any  training  in  com- 
binations with  ones. 

(12)  In  actual  instruction  many  teachers  do  not  drill  one  type 
of  combination  any  more  than  another.     The  additional  drill 
comes  later  when  the  child  fails  or  gets  confused.    Additional 

drill  is  used  as  cure  rather  than  as  preven- 
tion of  mistake.     Of  the  four  types  given, 
4  +  5  =   9        which  is  the  easiest  for  children  ?     Which 
3  +  7  =  10        the  hardest?    If  errors  are  more  frequent  in 
9  +  6=15        some  types  than  in  others,  is  this  due  to 
10  +  8  =  18        the  innate  difficulty  of  certain  types  or  to 
the  methods  of  teaching  them?     Do  chil- 
dren   add    from    large  to   small  numbers 
(9  +  5)  more  readily  than  from  small  to  large  numbers  (5+9 
-14)? 

(13)  Some  courses  of  study  require  that  a  combination  once 
learned    (5  +  7=12)    be    applied    immediately    to    the   higher 
decades   (15  +  7  =  22,  25  +  7  =  32,  etc.).     How  much  superior 
in  column  addition  is  a  class  thus  trained  to  one  not  so  trained  ? 
Is  it  necessary  to  apply  all  combinations  learned  in  this  way? 
May  it  not  be  that  the  general  idea  of  application  is  soon  acquired 
with  the  first  few  combinations  and  that  special  drill  is  not 
required  thereafter?     Are  there  certain    combinations    where 
special  drill  must  be  insured  always   (5  +6  =  eleven,  15  +  6  = 
twenty-one)  because  the  sound  regularity  is  interfered  with  ?  Or 
may  a  strictly  written  presentation  do  away  with  the  necessity 
of  special  drill  even  here? 

(14)  Is  there  any  increase  of  efficiency  in  drilling  on  combina- 
tions in  columns  as  soon  as  possible?     As  soon  as  the  com- 
binations that  add  up  7  are  learned  is  there  a  special  advantage 
in  immediately  giving  the  child  such  columns  in  application  as 
the  following: 


Details  for  Experiment  71 

2  4     24 

3  i     3i 

2      2       22 

7    7    77 

(15)  In  some  texts  and  courses  of  study  the  addition  combina- 
tions are  presented  in  the  order  of  the  sizes  of  the 

sums,    thus    2+2  =  4,   2  +  3  =  5,   etc.     In    others  6 
the  combinations  are  presented,  regardless  of  the  size  3 
of  the  numbers  involved,  so  as  to  immediately  fit  into  6 
certain  drill  columns  already  prepared.    Thus  the  an-  4 
nexed  column  would  require  the  following  combina- 
tions    (beginning    from    the    bottom),    4  +  6=10,  19 
0  +  3  =  3,    and    3  +  6  =  9.      What    is    the    relative 
worth  of  these  two  methods? 

(16)  In  column  addition,  where  carrying  is  involved,  some 

rationalize  the  process,  and  others  teach  it  mechan- 

23          ically  as  a  mere  bit  of  habituation.     In  the  case 

47          here  given,  some  would  add  each  column   sepa- 

36          rately,  taking  a  second  total  of  the  partial  sums. 

Others  would  merely  "  put  down  the  six  and  add 

1 6          one  to  the  next  column,"  writing  down  only  the 

9  complete  sum.     Which  is  superior,  in  that  it  will 

result  in  accurate  and  rapid  column  addition  in 

106          the  shortest  space  of  time? 

The  preceding  treatment  of  addition  will  suggest 
similar  problems  as  more  or  less  recurring  in  subtraction,  e.  g., 
whether  there  is  any  gain  in  teaching  5  —  3  =  2  immediately 
after  learning  that  5  —  2  =  3,  etc. 

(17)  Is  there  any  advantage    in    the    so-called    "Austrian" 
method  of  subtraction  by  addition,  over  the  method  of  subtracting 
through  specially  learned  subtraction  combinations?    If  so,  how 

much  considering  that  subtraction  and  addition 

4         10         (i)    mean   different  concrete   situations,    (2) 

+  6    —   6        use  a  different  written  algorism,  (3)  use  the 

same  oral  form  ("6  and  4  are  10"),  and  (4) 

10          4        employ  the  same  memorization   (6  and  4  are 

10)  ?    What  extent  of  school  energy  is  saved, 

if  any?    Is  the  method  of  subtraction  from  the  next  digit  in  the 


72  The  Teaching  of  Arithmetic 

top  number  superior  or  inferior  to  the  method  of  adding  to  the 
next  digit  in  the  bottom  number  ?  How  is  it  when  these  methods 
are  applied  to  the  addition-method  of  subtraction?  to  the  "  old  " 
subtraction-combination  method  ? 

(18)  Do  children  make  fewer  errors  when  they 

are  formally  taught  to  make  a  preliminary  inspec-  389 

tion   of   subtraction   examples   before   proceeding      —421 
to  manipulate  specific  combinations?  as  when  the 
number    cannot    be    subtracted;     as     when     the           345 
answer    is    zero.      (See  the  two   examples   here      — 345 
given.) 

(19)  Do  children  make  fewer  errors  and  manifest  less  con- 
fusion where  they  are  formally  taught  to  handle  the  zero  diffi- 
culties prior  to  being  confronted  with  them  in  column  subtrac- 
tion?   As  in  the  type  cases  given  below: 

(a)     867          (b)     867          (c)     867          (d)     870 
—467  —400  —  32  —650 


400                   467  835  220 
Where  borrowing  from  top? 

(e)     128           (f)     602  (g)     612  (h)     612 

—  76                —237  —318  —308 


52                    365  294  304 
Where  adding  to  bottom? 

(0     834           (j)     834  (k)     804  (1)     814 

— 406                — 496  — 496  — 406 

428  338  308  408 

In  subtraction,  what  preparation  is  needed  in  a  command  of 
zero  combinations  to  perform  the  column  subtraction?  (Note 
each  case  given  above.)  Is  there  some  general  mode  of  handling 
these  zeros  that  will  not  require  a  mastery  of  it  in  connection 
with  each  number  it  may  be  combined  with?  How  does  the 
above  apply  to  the  combinations  with  ones?  Where  a  one  is 
involved,  is  it  merely  counting  downwards  or  backwards?  Or 
is  the  subtraction  of  I  exactly  like  the  subtraction  of  3  or  4  or 
any  other  number? 

(20)  What  are  the  basic  combinations  required  to  perform 
any  given  column  subtraction?  In  what  form  may  they  be  best 


Details  for  Experiment  73 

mastered?  Are  zero  subtractions  (6  —  0  =  6)  and  subtractions 
with  one  (6—1  =  5)  to  be  included  or  omitted?  Are  the  re- 
verses to  be  taught  as  basic?  (7  —  2  =  5  and  7  —  5  =  2.)  Are 
subtraction  combinations  of  varying  difficulty?  Which  are  the 
most  difficult,  as  shown  by  children's  errors? 

(21)  Will  children  have  less  difficulty,  with  fewer  ensuing 
errors,  if  they  approach  column  subtraction  through  a  series  of 
graded  types  of  difficulty?  Consider  in  the  following: 

(a)     No  borrowing. 

1498 
—  964 


534 
(b)     Borrowing  each  time  save  the  last. 

8431 
—5987 


2444 
(c)     Borrowing  alternately. 

8431 
—2917 


5514 

In  what  order  should  types  (b)  and  (c)  be  given? 

How  rapidly  may  a  child  advance  from  two  or  three  figures 
to  seven  or  eight?  What  new  difficulties  present  themselves  in 
such  an  extension  of  figures? 

(22)  Are  the  first  series  of  multiplication  combinations  best 
presented  (i)  by  the  use  of  objects  grouped  and  counted,  or 
(2)  by  the  use  of  column  addition?  Or  are  these  two  methods 
best  used  as  supplementary  to  each  other?  Are  the  combinations 
with  zeros  (6X0  =  0)  and  the  combinations  with  ones 
(6  X  i  =  6)  best  taught  in  the  tables,  or  later,  in  con- 
nection with  their  actual  use  in  column  multiplication?  Is 
there  a  gain  in  teaching  the  reverses  in  connection  with  the  com- 
binations to  which  they  are  related,  exactly  as  with  the  addition 
combinations,  thus  6X3=18  immediately  after  3X6=18? 
What  gradation  of  steps  is  most  economical  and  efficient  in 
proceeding  from  combinations  to  their  application  in  column 
multiplication?  Since  partial  products  represent  but  stages  in 
calculation  do  they  need  to  be  understood  as  to  their  placing,  or 


74  The  Teaching  of  Arithmetic 

should  their  placing  be  taught  as  a  mechanical  process  through 
habit  formation?  Is  it  economical  to  allow  zeros  to  be  recorded 
which  later  will  be  abandoned?  as  in  multiplying  by  206?  What 
special  drill  on  zero  difficulties  (and  on  the  manipulation  of  ones) 
is  required  in  connection  with  their  handling  in  column  multipli- 
cation? How  can  this  be  best  provided? 

(24)  Is  there  any  need  for  division  tables  of  combinations? 
May  not  the  multiplication  tables  be  used  for  division,  precisely 
as  the  addition  combinations   are  used  for  subtraction?     For 
example,  from  "  3  2's  are  6,"  may  we  not  step  to  the  case  of  2)6? 
"  How  many  2's  are  6?  "    "3  2?s  are  6."    Here  the  identification 
is  through  a  common  oral  form  of  expression.    Is  there  any  need 
to  show  that  a  specific  written  form  or  algorism  in  multiplication 
is  the  equivalent  of  another  one  in  division,  since  this  is  not  done 
in  subtraction  by  addition? 

(25)  In   column   addition,   column   subtraction,    and   column 
multiplication  the  fundamental  combination  is  generally  obvious 
in  the  process  of  manipulation.     Is    the    division    combination 
equally  obvious  in  long  and  short  division?    Does  this  require 
special  treatment? 

(26)  Will  children  do  long  and  short  divisions  more  efficiently 
if  drill  in  division  with  a  remainder  (after  12-^-3  =  4,  learning 
13-4-3  =  4,  with  i  remainder,  and  14  -r-  3  =  4,  with  2  remainder) 
is  inserted  between  the  learning  of  the  combinations  and  their 
application  to  long  and  short  divisions?     Since  the  largest  diffi- 
culties seem  to  occur  in  connection  with  long  division,  and  since 
division  by  a  one-figure  divisor  must  precede  division  by  a  num- 
ber of  two  figures,  shall  division  by  a  one-figure  number  be  first 
taught  in  its  complete  form?     Shall  division  by  a  one-figure 
number  be  abridged  to  "  short  division  "  before    or    after    the 
development  of  division  with  a  divisor  of  two  figures?      Does 
the  distinction  between  partition  and  measuring  have  any  relation 
whatsoever  to  skill  in  manipulation?    What  is  the  worth  of  the 
distinction  in  interpreting  problems  and  applying  calculations? 

(27)  How  many  special  types  of  zero  difficulties  need  be  antici- 
pated and  carefully  drilled  upon  before  children  are  allowed  to 
attack  examples  where  zero  difficulties  are  likely  to  confront 
them?    Is  there  any  special  order  in  which  these  zero  difficulties 
should  be  attacked  for  purposes  of  economical  mastery?    What 


Details  for  Experiment  75 

special  training  in  the  handling  of  ones  should  be  provided  for, 
especially  if  the  one  combinations  in  the  tables  are  omitted? 

(28)  The  above  list  deals  entirely  with  investigations  in  teach- 
ing which  are  mainly  psychological.  There  is  another  large 
series  of  investigations  as  to  the  materials  required  in  the  various 
courses  of  study.  These  are  sociological.  In  such  investigations 
one  would  make  such  inquiries  as  the  following:  What  is  the 
demand  for  square  root  in  ordinary  business  occupations  ?  What 
types  of  fraction  examples  are  called  for  frequently,  and  what 
infrequently?  Which  types  of  reasoning  combinations  are  most 
used  ?  Do  we  "  add  and  multiply ''  within  the  same  problem, 
more  frequently  than  we  "  multiply  and  add  ?  "  Upon  such  social 
investigations  should  the  selection  and  the  omission,  the  emphasis 
and  the  subordination  of  specific  topics  be  determined.  Then 
will  our  courses  of  study  represent  the  highest  social  efficiency. 

BIBLIOGRAPHY:  Teachers  should  consult,  on  this  topic,  Pro- 
fessor Suzzallo's  work  already  cited. 


CHAPTER  XV 
THE  WORK  OF  THE  FIRST  SCHOOL  YEAR1 

The  first  question  that  naturally  arises  in  connection  with  the 
arithmetic  of  the  first  grade  is  as  to  whether  or  not  the  subject 
has  any  place  there  at  all.  For  several  years  past  there  has  been 
in  this  country  a  propaganda  in  favor  of  excluding  it  as  a  topic 
from  the  first  grade  and  even  from  the  second.  Like  all  such 
efforts,  the  history  of  which  is  not  generally  known,  the  very 
novelty  of  the  suggestion,  to  many  teachers,  is  sufficient  to  create 
a  following.  It  is  well  to  consider  briefly  the  reasons  for  and 
against  such  a  suggestion,  and  to  attempt  to  weigh  these  reasons 
fairly  before  attempting  any  decision. 

In  favor  of  having  no  arithmetic  as  such  in  the  first  grade  it 
is  argued  that  the  spirit  of  the  kindergarten  should  extend  farther, 
perhaps  even  through  all  of  the  primary  grades ;  that  number 
work  should  come  in  wherever  there  is  need  for  it,  all  learning 
being  made  attractive  and  natural,  and  education  appearing  to 
the  child  as  a  unit  instead  of  being  made  up  of  scattered  frag- 
ments. Such  a  theory  has  much  to  commend  it,  not  only  in  the 
primary  school  but  everywhere  else.  Opposed  to  it  is  the  rather 
widespread  idea  that  most  kindergarten  work  is  superficial  in 
aim  and  unfortunate  in  result;  that  children  who  have  had  this 
training  are  wanting  in  even  the  little  seriousness  of  purpose  that 
they  should  have,  that  they  have  no  power  of  application,  that 
they  have  been  "  coddled  "  mentally  into  a  state  that  requires 
constant  amusement  as  the  condition  to  doing  anything.  The 
dispassionate  onlooker  in  this  old  controversy  probably  feels  that 
there  is  truth  in  both  lines  of  argument,  and  that  mutual  good 
has  been  the  result.  Ancient  education  was  a  dreary  thing,  and 

1  It  is  impossible  in  the  space  allowed  to  enter  very  fully  into  details 
as  to  the  work  of  the  various  grades.  Teachers  who  desire  such  details 
may  consult  the  author's  Handbook  to  Arithmetics  (Boston,  1905).  All 
that  can  be  done  in  this  article  is  to  give  a  brief  survey  of  some  of  the 
most  important  topics  relating  to  the  various  grades. 

76 


The  Work  of  the  First  School  Year  77 

to  the  spirit  of  the  kindergarten,  although  not  to  extreme 
Frobelism,  we  are  indebted  for  the  brighter  spirit  of  the  modern 
school.  On  the  other  hand,  to  make  children  self-reliant,  inde- 
pendent in  thinking,  conscious  of  working  for  a  purpose,  demands 
more  thought  than  seems  to  pervade  the  ordinary  kindergarten. 

Now  as  to  arithmetic  in  the  first  grade:  Shall  we  leave  it  to 
the  ordinary  teacher  to  bring  in  incidentally  such  number  work 
as  he  wishes,  or  shall  we  lay  down  a  definite  amount  of  work  to 
be  accomplished  and  assign  a  certain  amount  of  time  to  it  ?  And 
in  answering  these  questions,  are  we  bearing  in  mind  the  average 
primary  teacher  throughout  the  whole  country?  Are  we  also 
bearing  in  mind  that  arithmetic  was  never  taught  to  children  just 
entering  school  until  about  a  century  ago,  and  that  it  was  largely 
due  to  Pestalozzi's  influence  that  the  subject  was  ever  placed  in 
the  first  grade?  When,  therefore,  we  advocate  having  no  arith- 
metic in  the  first  grade,  we  are  going  back  a  hundred  years  or 
so,  which  may  be  all  right,  but  which  is  not  a  new  proposition 
by  any  means. 

Having  thus  laid  a  foundation  for  an  answer  to  the  question, 
it  is  proper  to  proceed  dogmatically,  leaving  the  final  reply  to  the 
reader.  Not  to  put  arithmetic  as  a  topic  in  the  first  grade  is  to 
make  sure  that  it  will  not  be  seriously  or  systematically  taught 
in  nine-tenths  of  the  schools  of  the  country.  The  average  teacher, 
not  in  the  cities  merely  but  throughout  the  country  generally,  will 
simply  touch  upon  it  in  the  most  perfunctory  way.  Whatever  of 
scientific  statistics  we  have  show  that  this  is  true,  and  that 
children  so  taught  are  not,  when  they  enter  the  intermediate 
grades,  as  well  prepared  in  arithmetic  as  those  who  have  studied 
the  subject  as  a  topic  from  the  first  grade  on. 

Furthermore,  while  it  is  true  that  the  essential  part  of  arith- 
metic can  be  taught  in  about  three  years,  it  cannot,  for  psycho- 
logical reasons,  be  as  well  retained  if  taught  for  only  a  short 
period.  The  individual  needs  prolonged  experience  with  number 
facts  to  impress  them  thoroughly  on  the  mind.  We  can,  for 
example,  teach  the  metric  system  in  an  hour  to  any  one  of  fair 
intelligence,  but  for  one  to  retain  it  requires  long  experience 
in  its  use. 

But  more  important  than  all  else  is  the  consideration  of  the 
child's  tastes  and  needs.  Has  he  such  a  taste  for  number  as 


78  The  Teaching  of  Arithmetic 

shows  him  mentally  capable  of  studying  the  subject  at  the  age 
of  six,  and  are  his  needs  such  as  to  make  it  advisable  for  him 
to  do  so  ?  There  can  be  no  doubt  as  to  the  answer.  He  takes  as 
much  delight  in  counting  and  in  other  simple  number  work  in  the 
first  grade  as  in  anything  else  that  the  school  brings  to  him,  and 
he  makes  quite  as  much  use  of  it  in  his  games,  his  "  playing 
store,"  his  simple  purchases,  his  reading,  and  his  understanding 
of  the  conversation  of  the  home  and  the  playground,  as  he  does 
of  anything  else  he  learns.  If  we  could  be  certain  that  in  the 
incidental  teaching  that  is  so  often  advocated  he  would  have  these 
tastes  and  needs  fully  satisfied,  then  arithmetic  as  a  topic  might 
be  omitted  from  the  first  or  any  other  grade;  but  since  we  are 
pretty  sure  that  this  will  not  be  accomplished  in  the  average 
school,  then  it  is  our  duty  to  advocate  a  definite  allotment  of  time 
and  of  work  to  the  subject  in  every  grade  from  the  first  through 
the  eighth. 

This  being  so,  what  should  tnis  allotment  of  work  be?  Of 
course  there  is  no  general  answer  for  the  whole  country.  In 
some  schools  there  are  many  foreign  born  pupils  who  are  unable 
to  speak  English  when  they  enter  and  therefore  the  first  year's 
work  must  be  devoted  largely  to  acquiring  the  language.  In 
other  schools  the  children  come  from  homes  where  they  have 
already  been  taught  by  governesses  and  are  considerably  advanced 
over  the  average.  In  general,  however,  the  course  here  laid 
down  may  be  considered  a  fair  average  for  the  ordinary  American 
school. 

The  Leading  Mathematical  Feature.  The  introduction  to  the 
addition  table,  this  being  at  the  same  time  the  simplest  and  the 
most  important  operation  in  arithmetic.  It  is  not  advisable  to 
use  a  text-book  in  this  year,  on  account  of  the  children's  in- 
ability to  read. 

Number  Space.  It  has  been  found  best  both  from  the  stand- 
point of  mental  ability  and  of  needs  of  the  children  to  set  a 
different  limit  to  the  numbers  used  in  counting  and  in  the  opera- 
tions. Children  like  to  and  need  to  count  numbers  that  are  larger 
than  those  used  in  operations.  For  reading  and  writing  numbers, 
therefore,  they  may  profitably  go  as  far  as  100,  meeting  these 
numbers  in  the  paging  of  books,  the  numbering  of  houses,  the 
playing  of  games,  and  the  counting  of  various  objects.  For  the 


The  Work  of  the  First  School  Year  79 

operations,  however,  it  is  sufficient  if  they  go  as  far  as  12. 
Indeed,  10  would  make  a  good  limit  were  it  not  for  the  fact  that 
in  measuring  they  so  often  use  12  inches. 

Addition.  The  addition  tables  should  be  learned  at  least  as 
far  as  sums  of  10  or  12.  Some  prefer  to  go  as  far  as  9  +  4  =  13, 
but  it  is  immaterial  so  long  as  the  children  know  the  table  through 
9's  before  the  text-book  is  used, — ordinarily  the  middle  or  the 
end  of  Grade  II.  Appropriate  combinations  for  the  first  year 
may,  therefore,  be  taken  as  follows: 

123456789 
I     I     I     I     I     I     I     I     I 


23456789  10 
12345678 

22222222 

3456789  10 

1234567 

3333333 

4  5  6  7  8  9  10 

123456       12345 
444444      55555 

56789  10      6789  10 

1234     123 
6666     777 

7  8  9  10     8  9  10 

12       I 

88     9 

9  10    10 

This  arrangement  makes  the  sum    the    basis    for    selection. 
Many  prefer,  however,  to  proceed  to  master  the  table  of  I's,  2's, 


8o  The  Teaching  of  Arithmetic 

3's,  and  4's,  as  mentioned  above,  thus  giving  the  following  com- 
binations : 

123456789  10 
iiiiiiiiii 


1234 

2222 


1234 

3333 
4567 

1234 

4444 

5678 

It  is  not  a  matter  of  great  importance  which  of  these  two 
arrangements  is  adopted  in  any  given  school  system,  at  least  so 
far  as  we  are  able  to  judge  from  any  scientific  investigations  thus 
far  made.  The  great  thing  is  that  the  complete  table  shall  be 
known  to  10  +  10  by  the  end  of  the  second  year. 

Subtraction.  Every  fact  learned  in  addition  should,  judging 
from  general  experience,  carry  with  it  the  inverse  subtraction 
case.  That  is,  the  question  "3  +  2  equals  what  number  ?  "  should 
carry  with  it  the  questions  "  3  +  what  number  equals  5  ?  "  and 
"  2  +  what  number  equals  5  ?  "  or,  if  preferred,  "5  —  3  equals 
what  number?"  and  "5  —  2  equals  what  number?" 

Multiplication.  Little  attention  should  be  given  to  this  subject 
in  the  first  grade.  The  idea  that  2  +  2  +  2  may  be  spoken  of  as 
3  times  2,  and  the  incidental  use  of  the  word  "  times  "  in  other 
simple  number  relations  is  desirable. 

Division.  Since  multiplication  is  not  taken  as  a  topic,  its 
inverse  (division)  has  no  place,  save  as  it  appears  in  the  fractions 
mentioned  below. 


6 

7 

8 

9 

IO 

ii 

5 

6 

7 

8 

9 

IO 

2 

2 

2 

2 

2 

2 

7 

8 

9 

IO 

II 

12 

5 

6 

7 

8 

9 

IO 

3 

3 

3 

3 

3 

3 

8 

9 

IO 

ii 

12 

13 

5 

6 

7 

8 

9 

IO 

4 

4 

4 

4 

4 

4 

9 

IO 

ii 

12 

13 

14 

The  Work  of  the  First  School   Year  81 

Fractions.  Children  so  often  hear  about  the  fractions  l/2,  %, 
and  y3,  that  these  ideas  and  forms  may  profitably  be  introduced 
at  this  time,  although  y3  may  be  postponed  to  the  next  grade. 
The  statement  that  half  the  class  may  go  to  the  blackboard,  the 
idea  of  %  of  a  dollar,  and  that  of  l/$  of  a  yard,  are  all  common  in 
the  first  year.  In  the  introduction  of  these  ideas  and  symbols  it 
is  well  to  avoid  extremes  that  will  militate  against  the  child's 
future  progress,  such  as  the  extreme  of  the  ratio  method,  for 
example.  We  should  remember  that  a  fraction,  say  y>,  is  com- 
monly used  in  three  distinct  ways,  and,  that  it  is  our  duty  to  see 
that,  little  by  little,  all  these  become  familiar  to  the  child.  These 
ways  are  as  follows:  (i)  l/2  of  a  single  object,  the  most  natural 
idea  of  all,  the  breaking  of  an  object  into  2  equal  parts;  (2)  ^ 
as  large,  as  where  a  6-inch  stick  is  l/2  as  long  as  a  foot  rule, — 
not  half  of  it,  but  half  as  long  as  it  is;  this  is  essentially  the 
ratio  notion,  and  it  is  necessary  to  the  child's  stock  of  knowledge, 
but  it  is  not  necessary  to  make  it  hard  by  talking  about  ratios  at 
this  time;  (3)  */2  of  a  group  of  objects,  as  in  the  case  of  l/2  of 
ten  children. 

Denominate  Numbers.  Children  in  this  grade  should  learn 
the  use  of  actual  measures.  They  should  know  that  12  in.  =  i  ft., 
3  ft.  =  i  yd.,  and  should  employ  this  knowledge  in  making  meas- 
urements. They  should  know  the  cent,  5-cent  piece,  dime,  and 
the  dollar  as  10  times  (or  even  100  cents),  and  should  use  toy 
money  in  playing  store.  They  should  know  the  pint  and  quart, 
and  use  these  in  measuring  water  or  other  convenient  substance. 
Other  terms  such  as  pound,  week,  minute,  mile,  and  gallon  may 
be  used  incidentally,  but  they  should  not  be,  learned  in  tables, 
at  present. 

Objects.  It  is  important  to  use  objects  freely  wherever  they 
assist  in  understanding  number  relations,  but  it  is  equally  impor- 
tant to  abandon  them  as  soon  as  they  have  served  their  purpose. 
The  continued  use  of  any  particular  set  of  objects  (blocks,  disks, 
measures,  picture  cards,  etc.)  is  tiresome  and  narrowing.  Pesta- 
lozzi  was  wiser  than  many  of  his  successors  when  he  used 
anything  that  came  to  hand  to  illustrate  most  of  his  number 
work.  To  continue  to  use  objects  after  they  have  ceased  to  be 
necessary  is  like  always  encouraging  a  child  to  ride  in  a  baby 
carriage. 


82  The  Teaching  of  Arithmetic 

Symbols.  It  cannot  be  too  strongly  impressed  upon  teachers 
that  the  symbols  that  children  should  visualize  are  those  that  they 
will  need  in  practical  calculation.  Thus  it  is  much  better  to 
drill  upon  the  annexed  forms  than  upon 
6  9  9  6  +  3  =  9,  9  —  6  =  3,  9  —  3  =  6,  since 
+  3  —  6  —  3  the  latter  are  never  used  in  calculation. 
—  —  — ,  For  ease  in  printing  and  writing,  symbols 
936  like  6  +  3  =  9  have  their  important 
place,  but  the  eye  should  become  accus- 
tomed to  the  perpendicular  arrangement  so  as  to  catch  number 
combinations  as  it  must  do  when  we  come  to  actual  addition. 

Technical  Expressions.  While  it  is  proper  to  begin  by  reading 
6  +  2  "  six  and  two  "  and  8  —  6  "  eight  less  six,"  the  words 
"  plus  "  and  "  minus  "  should  soon  enter  into  the  vocabulary  of 
the  child  as  part  of  the  technical  language  of  the  subject.  It  is 
proper  to  call  a  cat  a  "  pussy  "  for  a  while,  and  a  horse  a  "  pony," 
but  the  time  soon  comes  for  "  cat "  and  "  horse," — and  so  for 
the  technical  expressions  in  arithmetic. 

Nature  of  the  Problems.  In  this  grade  problems  of  play,  of 
the  simplest  home  purchases,  and  of  interesting  measures  should 
dominate.  In  general,  for  all  grades,  the  oral  problems  should 
have  a  local  color,  relating  to  real  things  that  the  children  know 
about.  The  building  of  a  house  near  the  school,  the  repairing  of 
a  street,  the  cost  of  school  supplies — these  and  hundreds  of  simi- 
lar ideas  may  properly  suggest  problems  adaptable  to  every  school 
year.  It  is  the  business  of  the  text-book  in  the  grades  where  it 
is  used  to  furnish  a  large  amount  of  suggestive  written  work, 
but  it  can  never  furnish  all  the  oral  work  needed  nor  can  it 
meet  all  local  conditions. 

As  a  specimen  of  the  early  work  in  this  grade  the  following 
oral  exercise  is  submitted:1 

1.  How  many  inches  wide  is  the  window  pane? 

2.  How  many  feet  long  is  your  desk,  and  how  many  inches 

over? 

3.  How  many  feet  and  inches  from  the  floor  to  the  bottom  of 

the  blackboard? 


1  These  and  other  similar  sets  of  problems  used  in  this  article  are  taken 
from  other  works  of  the  author. 


The  Work  of  the  First  School  Year  83 

4.  Stepping  as  you  usually  do  in  walking,  find  how  many 

paces  in  the  length  of  the  room. 

5.  How  many  paces  wide  do  you  think  the  room  is?     Pace 

the  width  and  see  if  you  are  right. 

6.  How  tall  do  you  think  you  are?     Measure.     How  many 

feet,  and  how  many  inches  over? 

7.  How  many  inches  from  the  lower  left-hand  corner  of  this 

page  to  the  upper  right-hand  corner? 

8.  How  wide  do  you  think  the  door  is?     Measure.     How 

many  feet,  and  how  many  inches  over? 

Such  problems  suggest  measurements  of  genuine  interest  to 
the  pupil,  relating  as  they  do  to  his  immediate  surroundings. 
They  allow  for  the  actual  handling  of  the  measures  and  the  form- 
ing of  reasonably  accurate  judgments  concerning  distances. 

Abstract  Computation.  It  is  a  serious  error  to  neglect  abstract 
drill  work  in  arithmetic.  So  far  as  scientific  investigations  have 
shown,  pupils  who  have  been  trained  chiefly  in  concrete  problems 
to  the  exclusion  of  the  abstract  are  not  so  well  prepared  as  those 
in  whose  training  these  two  phases  of  arithmetic  are  fairly 
balanced.  Abstract  work  is  quite  as  interesting  as  concrete;  it 
is  a  game,  and  all  the  joy  of  the  game  element  in  education  may 
be  made  to  surround  it.  At  the  same  time  it  is  the  most  practical 
part  of  arithmetic,  since  most  of  the  numerical  problems  we  meet 
in  life  are  simplicity  itself  so  far  as  the  reasoning  goes;  they 
offer  difficulties  only  in  the  mechanical  calculations  involved,  and 
constantly  suggest  to  us  our  slowness  and  inaccuracy  in  the 
abstract  work  of  adding,  multiplying,  and  the  like.  In  the  first 
grade  this  work  is  largely  but  not  wholly  oral. 

Forms.  It  is  expected  that  children  in  this  grade  will  become 
familiar  with  the  names  of  the  common  solids  and  polygons 
needed  in  their  work.  For  example,  square,  rectangle,  triangle, 
oblong,  cube,  sphere,  cylinder,  pyramid,  prism,  and  similar  forms 
should  be  handled  and  their  names  should  be  known.  Paper 
cutting  and  folding  is  very  helpful  in  the  study  of  plane  figures 
and  in  the  work  with  fractions,  although  like  any  other  device, 
it  may  be  used  to  an  extreme  that  is  to  be  avoided. 

The  Time  Limit.  Even  in  the  first  grade,  and  still  more  in 
the  succeeding  years,  a  time  limit  should  be  set  on  all  number 


84  The  Teaching  of  Arithmetic 

work.  The  children  should  see  how  many  questions  they  can 
individually,  or  as  a  class,  or  as  half  of  the  class,  answer  in  a 
minute,  or  in  some  other  period  of  time.  Unless  this  is  done,  or 
some  similar  plan  is  adopted,  the  tendency  to  dawdle  over  the 
work  will  begin  to  crystallize  into  a  habit,  and  computation  will 
take  much  more  time  than  necessary.  It  is  also  to  be  observed 
that,  always  within  reasonable  limits,  rapid  calculation  contains 
less  errors  than  very  slow  work.  The  reason  is  apparent;  we 
concentrate  our  attention  more  completely,  and  other  thoughts  do 
not  take  our  minds  from  the  numerical  work. 

BIBLIOGRAPHY  :  The  author's  Handbook  to  Arithmetics,  p.  19  ; 
C.  A.  McMurry,  Special  Method  in  Arithmetic.  On  paper  fold- 
ing consult  Sundara  Row's  work,  Geometric  Paper  Folding 
(Open  Court  Publishing  Co.),  illustrated  by  photographs  taken 
by  the  author  of  the  present  work  a  few  years  ago.  This  work 
is  suggestive,  although  not  adapted  to  grade  work.  Consult 
also  Wentworth-Smith,  Stepping-Stones  in  Number,  Boston, 
1911. 


CHAPTER  XVI 
THE  WORK  OF  THE  SECOND  SCHOOL  YEAR 

Whether  or  not  arithmetic  has  a  definite  time  allotment  in  the 
first  grade,  it  usually  has  one  in  the  second,  although  some 
teachers  oppose  it  even  there.  The  argument  already  advanced 
holds  the  more  strongly  here,  especially  as,  in  many  schools, 
the  child  is  quite  prepared  to  use  a  text-book  by  the  middle  of 
this  year. 

The  Leading  Mathematical  Features.  In  schools  of  average 
advancement,  where  the  question  of  language  is  not  as  serious 
as  in  some  cities  in  the  East,  children  in  this  grade  may  be 
expected  to  complete  the  addition  tables  and  to  learn  the  multi- 
plication tables  to  10  X  5. 

Number  Space.  Children  will  now  take  an  interest  in  count- 
ing to  looo,  first  by  units  to  10,  then  by  ID'S  to  100,  then  com- 
pletely to  100,  then  by  loo's  to  1000,  and  finally  completely  to 
1000.  Their  operations  may  also  be  anywhere  within  this  space, 
although,  of  course,  most  of  their  results  will  involve  only  small 
numbers.  In  the  Roman  notation  the  limit  may  be  set  at  XII, 
this  sufficing  for  the  reading  of  time  and  for  the  chapter  num- 
bers of  their  books. 

Counting.  Without  going  to  an  extreme  in  counting  by 
various  numbers  where  no  definite  purpose  is  served,  there  is  a 
field  in  which  counting  is  very  advantageous.  To  count  by  2's 
from  2  to  10  and  from  I  to  n  has  the  pleasure  of  any  rhythmic 
sequence  and  at  the  same  time  gives  the  addition  table  of  2's,  and 
the  counting  by  2's  from  2  to  20  gives  the  corresponding  multi- 
plication table.  Similarly,  counting  by  3's  from  3  to  30  gives 
the  multiplication  table  of  3's,  while  the  further  counting  from 
i  and  2  to  13  and  14  gives  the  different  addition  combinations. 
The  exercise  is  interesting  to  children,  and  the  knowledge  secured 
in  this  way  is  more  than  one  would  at  first  think. 

Addition.     The  tables  should  be  completed  during  this  year, 

85 


86  The  Teaching  of  Arithmetic 

including  the  sums  of  any  two  one-figure  numbers.  There  are 
only  45  possible  combinations  of  numbers  below  10,  viz.:  I  +  I, 
1+2,  and  so  on  to  I  +  9 ;  2+2,  2  +  3,  and  so  on  to  2  +  9 ; 
3  +  3>  3  +  4>  an^  so  on  to  3  +  9 ;  and  similarly  for  the  others 
to  9  +  9,  besides  the  zero  combinations  referred  to  earlier  in 
this  paper.  It  is  better,  however,  to  continue  the  sums  to 
include  10, — a  simple  matter  but  one  that  is  often  helpful.  The 
addition  of  numbers  of  two  and  even  of  three  figures  each  may 
be  taken  during  this  year,  but  not  more  than  five  or  six  in  a 
column  should  be  used. 

Subtraction.  Subtraction  may  be  carried  far  enough  to  include 
numbers  of  three  figures  each.  The  method  to  be  employed  has 
already  been  discussed  in  Chapter  XI.  In  both  addition  and 
subtraction  there  should  be  an  effort  to  cultivate  the  habit  of 
rapidity,  although  never  to  the  exclusion  of  accuracy.  The  time 
limit  on  work,  mentioned  on  page  83,  should  be  employed  in  all 
written  work.  In  general  in  both  addition  and  subtraction  the 
full  form  should  be  employed  until  it  is  thoroughly  understood. 
For  example,  in  adding  247,  376,  and  85,  a  problem  that  must 
have  been  preceded  by  many  simpler  ones,  it  is  well  to  use  the 
first  of  the  following  forms  until  the  reasons  are  understood,  and 
then  to  adopt  the  second: 

247  247 

376  376 

85  85 


18  708 

190 
500 


Likewise,  if  the  addition  or  "Austrian  "  method  is  taken  for 
subtraction,  it  is  better  to  begin  a  problem  like  852  —  476  in  the 
full  form,  as  follows: 

852  =  800  +50  +  2 

476  =  400  +  70  +  6 

The  difference  between  these  is  the  same  if  we  add  10  to  each, 
and  also  100  to  each,  and  we  add  them  as  follows,  so  that  we  can 
easily  subtract  in  each  order: 

800+  150+  12 

500  +80+6 

300  +    70  +   6  =  376 


The  Work  of  the  Second  School  Year  87 

After  this  is  understood  we  may  proceed  to  the  ordinary 
arrangement. 

Multiplication.  The  multiplication  tables  may  be  learned  this 
year  as  far  as  10  X  5.  Some  schools  go  even  as  far  as  10  X  10, 
and  others  find  it  better  to  postpone  all  of  this  work  until  the 
third  grade.  Products  should  be  learned  both  ways,  i.  e.,  5X6 
and  6X5.  There  is  a  great  advantage  in  reciting  all  tables 
aloud,  and  even  in  chorus,  since  this  leads  to  a  tongue  and  ear 
memory  that  powerfully  aids  the  eye  memory  when  the  pupil 
needs  to  recall  a  number  fact.  Counting  enables  the  tables  to  be 
developed  in  a  rhythmic  fashion  that  is  pleasing  to  the  ear,  and 
shows  multiplication  by  integers  to  be  merely  an  abridged  addi- 
tion, that  is,  that  3  +  3  +  3  +  3  is  more  briefly  stated  as  4X3. 

Division.  The  multiplication  table  should  carry  with  it  the 
division  table.  This  need  not  be  developed  as  a  separate  feature 
but  may  be  treated  as  the  inverse  of  the  multiplication  table 
exactly  as  subtraction  is  the  inverse  of  addition.  The  fact  that 
4  X  6  =  24  should  bring  out  the  second  direct  fact  that  6X4 
=  24,  and  the  two  inverses,  24-1-6  =  4,  and  24-^-4  =  6.  These 
inverses  may  be  introduced  in  a  way  that  is  analogous  to  that 
followed  in  subtraction.  That  is  to  say,  after  learning  that 
4  +  5  —  9  we  ask,  "  What  number  added  to  5  equals  9  ?  "  "  What 
number  added  to  4  makes  9  ?  "  Similarly,  after  4  X  5  =  20  we 
ask,  "  What  number  multiplied  by  4  equals  20?  "  "  What  num- 
ber multiplied  by  5  equals  20?"  These  may  then  be  expressed 
as  20  •*-  4  =  5,  20  -4-  5  =  4. 

In  division  in  this  grade  we  also  have  an  illustration  of  the 
fact  that  the  full  form  should  precede  the  short  one.  A  child 
more  easily  grasps  the  idea  of  36  -f-  3  if  he  sees  the  first  of  these 
forms  before  he  comes  to  use  the  second: 

3)30  +  6  3)36 

10+2  12 

In  the  same  way,  when  he  comes  to  divide  36  by  2,  it  is  better 
to  begin  with  the  first  of  the  following  forms : 
2)20+16  2)36 

10+8  18 

Teachers  will  find  it  better  to  write  the  quotient  below  the 
dividend  in  short  division,  even  though  it  is  preferably  written 


88  The  Teaching  of  Arithmetic 

above  in  the  long  process.  There  is  no  advantage  in  trying  to 
change  the  habit  of  the  world  on  such  a  small  matter.1 

Fractions.  Children  know  the  meaning  of  l/2,  %,  and  often 
of  y$,  on  entering  this  grade.  If  y$  is  not  known  it  should  be 
introduced  and  ^,  Vc>  Vs  mav  a^so  ^e  added  to  the  list  at  this 
time,  although  many  successful  teachers  prefer  to  postpone  them 
until  Grade  III.  The  use  of  objective  work  is  imperative,  and 
it  is  better  to  take  various  simple  materials  than  to  confine  one's 
self  to  elaborate  fraction  disks  or  other  similar  devices.  Every 
school  has  cubes  to  work  with,  and  the  use  of  cubes,  paper  fold- 
ing, paper  cutting,  and  the  common  measures  is  recommended 
as  quite  sufficient. 

Denominate  Numbers.  The  denominations  already  learned  in 
Grade  I  should  be  frequently  used,  and  to  them  should  be  added 
the  relation  between  the  ounce  and  pound;  the  pint,  quart,  and 
gallon;  the  quart,  peck,  and  bushel;  the  reading  of  time  by  the 
clock,  and  the  current  dates.  The  idea  of  square  measure  (in 
square  inches)  is  introduced.  All  of  this  work  should  be  done 
with  the  measures  actually  in  hand  so  far  as  this  is  possible.  A 
table  of  denominate  numbers  means  very  little  unless  accompanied 
by  the  real  measures.  This  will  be  felt  by  any  American  grade 
teacher  who  teaches  the  metric  system  without  the  measures,  and 
who  tries  to  think  of  his  weight  in  kilos,  his  height  in  centi- 
meters, and  the  distance  to  his  home  in  kilometers. 

Symbols.  It  has  already  been  said  that  symbols  like  +,  — , 
X,  and  -T-  were  invented  for  algebra  and  have  only  recently  found 
place  as  symbols  of  operation  in  arithmetic.2  The  desire  to 
employ  them  has  led  many  teachers  to  use  long  chains  of  opera- 
tions that  are  never  seen  in  practical  life  and  which,  while  serving 
some  purpose  in  oral  work,  are  vicious  as  written  exercises.  For 
example,  2+4-^-2  +  5X6-^-3  +  3  is  a  kind  of  work  that  should 
never  appear  in  the  grades.  Arithmetically  it  is  easy  enough, 
and  the  answer  is  17,  but  there  is  no  use  in  puzzling  a  child 
to  remember  which  signs  have  the  preference  in  such  a  chain. 
This  is  a  small  technicality  of  algebra,  of  which  the  impor- 
tance is  much  overrated  even  there,  and  it  has  no  place  in 


1  See  the  author's  Handbook,  p.  27. 

2  It  is  true  that  +  and  —  were  first  used  in  Widman's  arithmetic  of 
1489,  but  not  as  symbols  of  operation.     See  my  Kara  Arithmetica. 


The  Work  of  the  Second  School  Year  89 

the  elementary  school.     With   respect   to   the  symbols  2  X  $3 

and  $3X2  there  is,  however,  a  reasonable  question,  since  there 

is  good  authority  for  each.     Modern  usage  favors  the  former 

because  we  more  naturally  say  "  2  times  3  dollars  " 

$3        than  "  3  dollars  multiplied  by  2,"  and  it  is  better  to 

2        read  from  left  to  right  as  in  an  ordinary  sentence.   It 

should  be  repeated,  however,  that  the  forms  which  the 

$6        child  needs  to  visualize  are  not  these  but  the  one  he 

will  meet  in  actual  computation,  as  here  shown. 

Objects.  It  is  here  repeated,  as  essential  to  a  discussion  of 
the  work  of  the  second  grade,  that  objects  are  necessary  in  devel- 
oping certain  number  relations,  but  that  they  should  be  discarded 
as  soon  as  the  result  is  attained.  Number  facts  must  be  memo- 
rized by  every  one,  and  objects  may  become  harmful  if  used 
too  often. 

Nature  of  the  Problems.  This  matter  begins  to  assume  con- 
siderable importance  in  this  grade,  and  it  has  been  already  dis- 
cussed in  Chapter  IV.  It  may  be  said  in  general,  however,  that 
several  of  our  recent  American  arithmetics  are  making  a  serious 
effort  to  improve  the  applications  of  the  subject,  adapting  them 
to  the  mental  powers  and  to  the  environment  of  the  pupils  in- 
stead of  offering  obsolete  material  of  no  practical  value  and  of 
little  interest. 

Necessity  for  Systematic  Reviews.  It  is  proper  at  this  time  to 
call  the  attention  of  teachers  to  the  matter  of  reviews, — not  those 
that  naturally  occur  from  time  to  time  during  the  year,  but  those 
that  should  deeply  concern  every  school  at  the  close  of  one  year 
and  at  the  opening  of  the  next  one.  Any  one  who  has  ever  had 
much  to  do  with  the  supervision  of  the  grade  work  in  arithmetic 
is  struck  by  the  general  complaint  that  children  are  never  pre- 
pared to  enter  any  particular  grade.  Every  teacher  seems  to  feel 
that  the  preceding  teacher  has  imposed  a  poorly  equipped  lot  of 
children  upon  her  own  grade  and  that  her  problem  is  therefore 
hopeless.  Now  if  this  were  only  an  occasional  complaint  the 
supervisor  might  well  be  worried,  but  he  soon  recognizes  it  as 
part  of  the  tradition  of  the  school,  and  pays  little  attention  to  it 
accordingly.  What  does  it  mean,  however,  and  how  should  we 
remedy  the  evil  if  evil  there  be? 

If  any  teacher  will  himself  learn,  let  us  say,  the  logarithms  of 
the  first  fifty  integers,  between  September  and  February,  how 


po  The  Teaching  of  Arithmetic 

many  will  he  know  in  June?  And  if  he  knows  them  all  in  June 
how  many  will  he  remember  at  the  end  of  the  summer  vacation  ? 
And  how  will  he  feel,  say  about  September  15,  if  some  one 
suddenly  asks  him  to  give  the  logarithm  of  37  to  six  decimal 
places,  telling  him,  if  he  fails,  that  he  must  have  been  pretty 
poorly  taught  the  year  before  ?  Now  this  is  a  fair  illustration  of 
the  mental  position  of  a  child  with  respect  to  the  multiplication 
table  when  he  enters  Grade  IV.  Psychologically  it  would  be 
strange  if  he  could  rapidly  and  accurately  give  every  product 
demanded ;  his  brain  cells  have  clogged  up  or  got  disarranged  or 
gone  through  some  similar  transformation  during  his  nine  or 
ten  weeks  of  careless  play.  What,  therefore,  is  the  teacher's 
duty?  There  are  two  things  to  do.  First,  at  the  close  of  each 
school  year,  in  June,  there  should  be  a  thorough  and  systematic 
review  of  those  number  facts  and  operations  that  are  the  funda- 
mental features  of  the  year's  work.  The  teacher  ought  to  be 
satisfied  that  each  child  leaves  the  grade  with  such  a  mental 
equipment  as  shall  leave  no  chance  of  fair  criticism.  His  respon- 
sibility then  ceases.  Second,  and  even  more  important,  at  the 
opening  of  each  school  year,  in  September,  there  should  again 
be  a  thorough  and  systematic  review  by  the  teacher  in  the  next 
grade,  of  these  same  features.  But  this  review  should  be  con- 
ducted in  the  most  sympathetic  spirit.  The  teacher  should  be 
surprised  if  the  children  have  not  forgotten  much  rather  than 
if  they  have  failed  to  remember  the  facts  perfectly.  He  should 
think  of  his  own  fifty  logarithms,  for  example,  and  the  review 
should  be  patiently  and  helpfully  extended  until  the  children's 
arithmetical  brain-cells  resume  their  former  state.  After  this 
has  been  done  in  the  spirit  mentioned,  and  after  the  teacher  has 
gone  into  the  next  higher  grade  for  a  day  to  see  'how  his  own 
pupils  of  the  preceding  year  are  standing  the  test,  then  he  may 
be  justified  in  complaining,  but  not  before. 

It  need  hardly  be  mentioned  that  there  are  few  more  severe 
tests  of  the  ingenuity  and  patience  of  a  teacher  than  are  found  in 
these  reviews.  The  "  edge  of  interest  "  is  already  worn  off  in  any 
review,  and  it  requires  all  the  tact  a  teacher  possesses  to  maintain 
the  enthusiasm  of  the  pupils  in  such  exercises.  The  result,  how- 
ever, is  well  worth  the  effort,  and  the  school  system  that  carries 
out  the  plan  will  have  less  of  complaint  and  more  or  sympathetic 
cooperation  than  would  at  first  be  thought  possible. 


CHAPTER  XVII 
THE  WORK  OF  THE  THIRD  SCHOOL  YEAR 

The  Preparation.  Since  the  text-book  is  placed  in  the  hands 
of  children  during  the  latter  part  of  the  second  school  year  or  at 
the  opening  of  the  third,  it  becomes  particularly  important  to 
have  a  systematic  review  of  the  work  of  Grades  I  and  II  at  the 
beginning  of  this  year.  The  text-books  usually  provide  for  this, 
and  by  their  help  these  important  things  are  accomplished:  (i) 
The  children's  memories  are  refreshed  as  to  the  essential  fea- 
tures of  the  preceding  year's  work,  viz.,  the  addition  table,  and 
the  multiplication  table  as  far  as  the  course  of  study  may  require. 
(2)  Children  are  "  rounded  up,"  brought  to  a  certain  somewhat 
uniform  standard,  so  that  all  can  begin  the  serious  use  of  the 
text-book  with  approximately  the  same  equipment.  (3)  The 
superior  capacity  or  the  defect  of  the  individual  has  an  opportu- 
nity to  show  itself  early,  allowing  for  such  advancement  or 
special  attention  as  the  case  demands.  In  other  words  the  "  lock- 
step  "  can  be  broken  without  the  usual  delay.  As  to  further  argu- 
ment for  this  autumnal  review  the  reader  may  refer  back  to 
Chapter  XVI. 

The  Leading  Mathematical  Features.  In  this  year  rapid 
written  work  is  an  important  feature.  The  oral  has  predomi- 
nated until  now,  but  in  Grade  III  the  operations  involve  larger 
numbers  than  before,  and  the  child  begins  to  acquire  the  habit 
of  writing  his  computations.  Multiplication  extends  to  two- 
figure  multipliers  and  long  division  is  begun.  The  most  useful 
tables  of  denominate  numbers  are  completed. 

Number  Space.  It  is  usually  considered  sufficient  if  the  child 
understands  numbers  to  10,000  in  this  grade,  although  he  may  be 
allowed  to  count  by  io,ooo's  to  100,000  or  even  farther.  Indeed, 
as  soon  as  he  understands  numbers  to  1,000  he  rather  enjoys 
showing  his  prowess  by  counting  by  i.ooo's  and  by  writing  large 
numbers.  Counting  always  extends  far  beyond  the  needs  of 


92  The  Teaching  of  Arithmetic 

computation — a  law  that  is  true  to-day  and  has  been  true  in  the 
historical  development  of  all  peoples.  In  the  writing  of  Roman 
numerals  there  is  no  particular  object  in  going  beyond  C  in  the 
first  half-year,  and  M  in  the  second  half.  It  must  be  borne  in 
mind  that  we  use  the  Roman  forms  chiefly  in  chapter  or  section 
numbers,  and  less  often  in  reading  dates,  so  that  all  writing  of 
very  large  numbers  by  this  system  in  an  obsolete  practice  and  a 
waste  of  time.  Indeed  it  is  not  strictly  a  Roman  system  any  more, 
so  much  have  we  changed  the  numerals  from  their  early  forms. 

Counting.  In  this  grade  the  counting  of  Grade  II  should  be 
continued,  including  the  6's,  7's,  S's,  9's,  and  lo's,  as  a  basis  for 
the  multiplication  tables  and  as  a  review  of  the  addition  combina- 
tions. There  is  no  need  of  counting  beyond  certain  definite  limits, 
however.  Thus  in  counting  by  2's  beginning  with  o,  we  have 
o,  2,  4,  6,  8,  10,  12,  14,  16,  18,  20.  This  suffices  for  the  multi- 
plication table  of  2's  and  even  the  last  half  of  this  is  merely 
a  repetition  of  the  first  half  with  10  added. 

The  Decimal  Point.  It  becomes  necessary  in  this  grade  to 
write  dollars  and  cents,  and  hence  forms  like  $10.75,  $25-IO>  &nd 
$32.02  are  given.  It  is  not  necessary  nor  even  desirable  that  the 
children  should  know  any  of  the  theory  of  decimal  fractions  at 
this  time.  The  decimal  point  should  be  looked  upon  by  them 
simply  as  separating  dollars  and  dimes,  and  it  will  give  no  trouble 
unless  the  teacher  confuses  the  class  by  the  ever-present  danger 
of  over-explaining. 

Forms.  It  is  usual  in  Grade  III  to  review  the  simple  geo- 
metric forms  already  learned,  such  as  the  triangle,  rectangle, 
cylinder,  and  sphere.  Formal  definitions  are,  however,  undesir- 
able. The  chief  thing  is  that  the  child  should  use  the  names 
correctly.  Some  little  paper-folding  may  well  be  introduced  as 
a  basis  for  simple  square  and  cubic  measure. 

Square  and  Cubic  Measure.  The  ideas  of  area  (square  inches 
or  square  feet)  and  volume  (cubic  inches  or  cubic  feet)  may 
enter  into  the  work  of  this  grade,  although  some  successful 
teachers  prefer  to  introduce  them  in  Grade  IV,  finishing  this 
work  in  Grade  V.  If  introduced  here,  they  are  of  course  treated 
objectively,  usually  with  paper-folding,  drawing,  or  inch  cubes  of 
wood.  There  is  hardly  any  trouble  with  this  work  unless  the 
teacher  enlarges  upon  its  difficulties.  If  there  is  accuracy  of 


The  Work  of  the  Third  School  Year  93 

language,  spoken  and  written,  from  the  beginning,  this  will  con- 
tinue; but  if  the  teacher  allows  expressions  like  "  3  inches  times 
3  inches  equals  9  square  inches,"  instead  of  "  3  times  3  square 
inches  equals  9  square  inches,"  there  will  be  produced  loose  habits 
of  thought  and  expression  that  will  lead  to  great  trouble. 

Devices  for  Fractions.     It  is  still  necessary  in  this  grade  to 
make  a  good  deal  of  use  of  objective  work  in  treating  fractions, 
and  to  make  the  work  largely  oral  dur- 
ing the  first  half  year.    There  is  also  an          2      3      4      5 
advantage  in  using  columns  of  figures  2345 

like  those  here  shown.    Here  it  is  very          2345 
easy  to  see  that  ^  of  8  is  two  2's,  or  4 ;          2      3      4      5 
that  y4  of  12  is  3 ;  that  ^4  of  16  is  three 
4's  or  12,  and  that  l/2  of  20  is  the  same          8     12     16    20 
as  V*  of  2O>  or  two  5's,  or  10.     From 
the  second  arrangement  it  is  easy  to  see  2 

that  2  is  y2  of  4,  y$  of  6,  J4  °*  8,  and  2      2 

YB  of  10;  that  4  is  ^  of  6,  y2  of  8;  that  222 

6  is  3/4  of  8  and  8/5  of  10,  and  so  on.          2222 
Devices  of  this  kind  add  both  to  the  in-          2222 
terest  in  and  clear  comprehension  of  the 
subject,  and  when  not  carried  to  an  ex-          4      6      8     10 
treme  are  valuable. 

Addition.    The  45  combinations  of  one-figure  numbers  should 
be  reviewed,  and  in  the  first  half  year  oral  work  of  the  types  of 
20  +  30,  25  +  30  should  be  taken,  to  be  followed  in  the  second 
half  year  by  cases  like  25  +  32  and  225  4-  32,  where  no  "  carry- 
ing "  is  involved.     Written  work  with  four-figure  numbers  in- 
cluding dollars  and  cents,  should  be  given,  but  long 
427        columns  of  figures  should  be  avoided  at  present. 
326  As  already  stated  there  is  an  advantage  in  intro- 

452         ducing  any  difficulty  in  operation  by  using  the  com- 
49        plete  form.    While,  for  example,  the  annexed  prob- 
lem in  addition  is  not  designed  as  an  introduction 
24        to  the  addition  of  three-figure  numbers,  it  illustrates 
130        what  is  meant  by  the  complete  form.    The  teacher 
noo        need  have  no  fear  that  children  cannot  easily  be 
brought  to  use  the  abridged  form ;  "  the  line  of  least 
1254        resistance "    will  bring  that  about,   while  on  the 


94  The  Teaching  of  Arithmetic 

score  of  a  clear  understanding  of  the  operation  this  complete 
form  is  far  superior  to  the  other.  It  should  also  be  mentioned 
that  the  pupil  should  at  this  early  stage  be  taught  to  recognize 
his  own  liability  to  error  and  to  do  what  every  computer  has  to 
do,  add  each  column  twice,  in  opposite  directions,  to  be  sure  of 
his  result, — to  "  check  "  it,  as  we  say. 

Subtraction.  This  subject  has  been  sufficiently  treated  under 
Chapter  XI.  The  extent  of  the  work  is  suggested  by  the  work 
in  addition,  and  of  the  various  methods  the  addition  or  "Aus- 
trian "  seems  at  present  to  be  the  best. 

Multiplication.     This,  with  division,  constitutes    the    special 
work  of  the  year,  addition  and  subtraction  offering  no  essentially 
new  difficulties.     In  the  first  half  year  it  is  customary  to  com- 
plete the  tables  through  10  X  10,  and  the  products  must  be  thor- 
oughly memorized  not  merely  in  tabular  form  but  when  called 
for  in  any  order.     The  plan  of  carrying  the  tables  to  12  X  12, 
while  necessary   in   England   on   account 
s  of  the  monetary  system  used  there,  has 

generally  been  discarded   in  America,   it 
being  felt  that  the  time  required  for  this 


24  =  3  X      8  extra  work  could  be  better  employed.    In 

270  =  3  X    90  the  first  half  year  multiplication  may  be 

carried  so  far  as  to  include  three-figure 
gQ4  =  o  x  208  multiplicands   and  one-figure  multipliers, 

208  and  the  work  may  at  first  be  arranged 

3  in  the  complete,   and  later  in  the  com- 

mon abridged  form  as  here  shown. 
Since  all  such  work  is  done  in  the  class- 
room where  the  teacher  can  supervise  it, 

there  should  be  a  time  limit  placed  upon  it,  to  the  end  that  habits 
of  rapidity  as  well  as  of  accuracy  should  be  acquired.     In  the 

second  half  year  the  work  may  usually 

298  be  extended  to  two-figure  multipliers,  in 

43  which  the  complete  form  should  again 

precede    the    common    abridgment,    as 

894  =   3  x  298  here  shown.     There  is  also  introduced 

11920  =  40X298  .      ...  ...  ,.     ,. 

in   this   year   such    multiplications    as 

12814  =  43X298  that  of  $2-75  by  7,  thus  preparing  the 

way   for  decimal   fractions.     The  lat- 


The  Work  of  the  Third  School  Year  95 

ter  are  not,  however,  treated  in  this  grade,  and  the  work  should 
not  be  made  difficult  by  any  unnecessary  theorizing  upon  this 
subject. 

Division.     In  this  year  oral  division  by  one-figure  divisors 
is  introduced  for  such  simple  cases  as  484  -j-  2,  484  -^-4,  481  -r-  2, 
etc.   Short  division  of  numbers  like  522  -r-  6, 
should  be  introduced  by  some  such  form  as 
)  522  the  annexed.     Such  separations  of  the  divi- 

6)480  +  42  dend  are  made  for  the  purpose  of  having  the 

8o+7  process  seen  in  its  simplest  form,  and  teach- 

ers should  write  problems  of  this  kind  on  the 
board  often  enough  to  make  sure  that  the 
process  is  understood.     The  children  should  not  be  required  to 
use  this  form,  but  should  get  to  the  practical  work  of  division 
as  soon  as  possible.     In  the  second  half  year  the  two-figure 

divisor  may  be  introduced,  but 

since  the  greatest  difficulty  in  di- 

75  vision  consists  in  the  estimating 

2I)I575  of  the    successive    quotient  fig- 

1470  =  70  X  21  .,    .          „   .  c 

ures,  it  is  well  to  confine    the 

IOr  divisors,  for  this  year,  to  those 

105  —    5  X  21  whose  unit  places  are  at  first  o, 

then   i,  and  finally  2.     As   an 


<^Q  .,-  early  form  for  long  division,  the 

2j)S7~77  annexed  algorism  is  suggested. 

6.30  =  21  X  $0.30  The  use  of  United  States  money 

again  brings  in  the  decimal  point 

1-47  so  naturally  that  the  difficulty  of 

1.47  =  21  >  decimal  fractions  is  much  dimin- 

ished when  that  topic  is  reached. 

The  full  form  that  may  properly  precede  the  common  abridg- 
ment is  here  set  forth. 

Scope  of  Work  with  Fractions.  The  pupil  is  now  able  to  use 
halves,  thirds,  fourths,  fifths,  sixths,  and  eighths,  or,  if  not,  this 
work  should  be  introduced  at  this  time.  Oral  addition  and  sub- 
traction of  fractions  with  a  common  denominator.  The  reduc- 
tion of  halves  to  fourths,  sixths,  and  eighths,  and  of  thirds  to 
sixths,  is  introduced  by  means  of  objects,  the  objects  being  dis- 
carded as  soon  as  they  have  served  their  purpose.  Fractional 


g6  The  Teaching  of  Arithmetic 

parts  of  numbers  of  three  figures  or  less,  these  being  selected  so 
as  to  be  multiples  of  the  denominator. 

Denominate  Numbers.  Here  as  in  other  grades  it  is  neces- 
sary to  review  and  frequently  •  use  the  tables  already  learned. 
The  table  of  square  units  is  often  introduced  and  extended  to 
the  square  yard,  although  it  may  be  postponed  a  year.  The 
gill  is  added  to  the  table  of  liquid  measure,  and  the  table  of 
time  is  completed.  Modern  teaching  finds  it  advisable  to  intro- 
duce the  units  of  measure  only  as  rapidly  as  the  child  develops 
the  need  for  them  and  can  therefore  understand  them.  In  all 
cases  it  is  desirable  to  have  the  measures  where  they  can  be 
seen  or  in  some  other  way  appreciated.  For  example,  when  the 
acre  is  introduced,  somewhat  later,  a  piece  of  land  near  the 
school,  approximately  an  acre  in  size,  should  be  shown  to  the 
class.  In  the  same  spirit  they  should  see  a  ton  of  hay  or  a 
ton  of  coal,  a  cord  of  wood  where  this  is  possible,  a  rod,  a 
gill  measure,  and  so  on.  It  is  very  important  that  the  great 
basal  units  used  by  our  people  should  be  visualized  by  the  chil- 
dren, so  that  bushel,  mile,  ton,  etc.,  shall  not  be  mere  words. 

Typical  Problems.  The  following  are  suggested  as  two  prac- 
tical sets  of  problems,  adapted  to  this  grade,  each  telling  a  story 
that  may  suggest  other  topics  for  original  work  by  the  class. 

Oral  Exercise 
Some  Home  Meals 

1.  The  coffee  for  our  breakfast  cost  6c.,  the  potatoes  4c.,  the 
meat  32c.,  and  the  bread  4c.    How  much  did  the  bread  and  meat 
cost?     How  much  did  all  the  food  cost? 

2.  The  oatmeal  for  a  breakfast  cost  8c.,  the  milk  4C.,  the 
fruit  ioc.,  the  rolls  and  butter  5c.,  and  the  eggs  8c.    How  much 
did  this  food  cost? 

3.  For  a  dinner  the  meat  cost  3Oc.,  the  vegetables  2oc.,  the 
dessert  2oc.,  the  coffee  I5c.,  and  the  other  food  I5c.    Find  the 
total  cost. 

4.  The  meals  for  a  small  family  cost  $1.70  on  one  day  and 
$2.20  on  another  day.    How  much  did  they  cost  for  these  two 
days? 


The  Work  of  the  Third  School  Year  97 

Written  Exercise 

The  toad  is  one  of  man's  best  friends.    One  toad  will  keep  a 
garden  of  800  sq.  ft.  free  from  harmful  insects. 

1.  At  this  rate,  how  many  toads  would  protect  from  insects 
a  garden  80  ft.  wide  and  100  feet  long? 

2.  The  eggs  of  4  toads  were  counted  and  found  to  be  7547, 
11,540,  7927,  and  9536.     How  many  were  there  in  all? 

3.  If  one  out  of  50  hatched,  how  many  hatched?     (Divide 
all  by  50.)     If  715  of  these  were  destroyed  by  other  animals, 
how  many  survived? 

4.  If  each  of  these  survivors  destroys   insects  that  would 
cause  $10  worth  of  damage,  how  much  are  they  all  worth  to  a 
village?      . 

BIBLIOGRAPHY:     The  author's  text-books  on  arithmetic,  and 
the  Wentworth-Smith  series,  set  forth  his  views  in  detail. 


CHAPTER  XVIII 
THE  WORK  OF  THE  FOURTH  SCHOOL  YEAR 

The  Leading  Mathematical  Features.  In  this  the  last  year  of 
the  primary  grades  it  is  well  to  feel  that  the  essentials  of  arith- 
metic have  all  been  touched  upon.  It  is,  therefore,  desirable  to 
review  the  four  fundamental  operations,  extending  the  multipli- 
cation and  division  work  to  include  three-figure  multipliers  and 
divisors.  The  common  business  fractions  should  also  be  in- 
cluded, with  simple  operations  as  far  as  multiplication. 

Number  Space.  In  the  first  half  year  the  numbers  may  extend 
to  100,000,  and  in  the  second  half  year  number  names  may  be 
given  as  high  as  a  billion.  The  operations,  however,  should  be 
confined  to  the  smaller  numbers  of  such  business  as  can  be 
appreciated  by  children  of  this  age. 

Counting.  The  prime  object  of  the  counting  exercises,  the 
developing  of  the  tables  of  addition  and  multiplication,  has  now 
been  accomplished,  except  when  it  is  desired  to  carry  the  multi- 
plication table  to  12  X  12.  In  that  case  the  counting  may  now 
be  continued  by  n's  to  132  and  by  I2's  to  144.  -Otherwise  the 
only  use  of  counting  in  this  grade  is  for  the  purpose  of  review. 

Addition  and  Subtraction.  There  should  be  much  rapid  oral 
work  with  numbers  like  the  following: 

7          17         37         37         47 

+    4    +    4    +    4+14+24 

II  21  41  51  71 

-    7    -    7    -  17    ~  27    -  47 

The  written  work  should  be  undertaken  with  the  aim  of  (i) 
accuracy,  secured  by  always  checking  the  result;  (2)  rapidity, 
secured  by  setting  a  time  limit  upon  all  work.  Children  should 
by  no  means  neglect  this  matter  of  checks,  since  it  is  used  in  all 
the  business  world.  Much; of  the  complaint  of  business  men, 

98 


The  Work  of  the  Fourth  School  Year  99 

that  boys  from  the  schools  are  always  inaccurate  in  arithmetic, 
would  be  obviated  if  pupils  were  always  required  to  check  their 
additions  by  adding  in  the  opposite  directions,  and  their  other 
results  in  some  appropriate  manner.  In  subtraction,  for  example, 
if  the  result  is  obtained  by  the  "Austrian  "  method  it  should  be 
checked  by  adding  it  to  the  subtrahend  in  the  opposite  direction. 

Multiplication  and  Division.  No  new  principles  are  involved 
here,  and  the  work  of  the  preceding  year  is  simply  extended  to 
include  larger  numbers.  In  some  schools  the  multiplication 
table  is  extended  to  12  X  12,  although  this  is  not  important 
enough  for  most  people  to  make  it  worth  the  while.  It  is  a  good 
plan,  however,  to  learn  all  products  less  than  50,  as  2  X  13, 
3  X  15,  4X  12,  and  so  on.  since  these  are  so  often  used  in  the 
purchases  of  the  household.  Even  a  child  ought  to  know  the 
cost  of  2  Ib.  of  meat  at  18  cents  a  pound,  without  using  pencil 
and  paper.  The  practical  checks  on  multiplication  and  division 
are  not  advantageously  discussed  as  early  as  this. 

Fractions.  Here  as  later  the  work  in  common  fractions  should 
be  confined  to  those  needed  in  ordinary  business,  and  at  present 
to  those  from  ]/2  to  %.  Of  course  there  is  no  objection  to  an 
occasional  example  with  denominators  of  two  or  three  figures, 
but  the  day  of  fractions  like  -$zfa  is  past,  decimal  fractions  having 
taken  the  place  of  all  such  forms.  Children  in  this  grade  should 
also  know  that  $l/2  =  50  cents,  and  $>4  —  25  cents.  The  opera- 
tions may  extend  as  far  as  easy  multiplications  of  an  integer  and 
a  fraction,  two  fractions,  or  an  integer  and  a  mixed  number. 
Unusual  forms  of  operation,  not  practical  in  business,  should  not 
be  given,  and  the  teacher  should  resist  all  temptation  to  depart 
from  this  principle  on  any  supposed  ground  of  mental  discipline. 

Decimal  Fractions.  A  brief  introduction  to  this  subject,  based 
on  the  work  already  given  in  United  States  money,  may  be  al- 
lowed in  this  grade,  although  the  serious  treatment  of  decimals 
belongs  later  in  the  course. 

Denominate  Numbers.  The  tables  needed  in  business  life  are 
completed  in  this  grade  by  adding  that  of  land  measure,  and 
completing  long  and  cubic  measure.  In  the  work  of  adding  and 
subtracting  compound  numbers  children  should  feel  that  there 


ioo  The  Teaching  of  Arithmetic 

is  no  principle  involved  that  is  not  found  in  integers.  For  ex- 
ample, consider  these  two  cases : 

37  3  ft-  7  in-  3  lb-     7  oz. 

25  2  ft.  5  in.  2  Ib.     5  oz. 

62  6  ft.  5  Ib.  12  oz. 

In  the  first,  because  7  +  5  =  12,  which  is  I  ten  and  2  units,  the 
I  ten  is  added  to  the  lo's.  In  the  second,  because  7  in.  +  5.  in.  — 
12  in.  or  i  ft.,  the  I  ft.  is  added  to  the  feet.  In  the  third,  because 
7.  oz.  +  5  oz.  =  12  oz.,  which  does  not  equal  a  pound,  it  is 
written  under  ounces.  In  every  case  the  principle  is  the  same, 
to  add  to  the  next  order  any  units  of  that  order  that  are  found. 
In  general  we  use  compound  numbers  of  only  two  denomina- 
tions, and  it  is  on  such  numbers  that  we  should  lay  the  emphasis. 
The  use  of  numbers  of  four  or  five  denominations  is  now 
obsolete,  and  there  is  not  enough  disciplinary  value  in  the  sub- 
ject to  warrant  using  them  instead  of  the  numbers  of  actual 
business. 

A!s  heretofore  mentioned,  there  should  be  an  effort  to  have 
children  visualize  the  standard  measures  of  our  country,  such 
as  the  acre,  mile,  ton,  and  bushel. 

Teachers  should  be  careful  at  this  time  that  slovenly  methods 
of  statement  do  not  become  habits.  Such  forms  as  the  follow- 
ing, for  example,  are  inexcusable: 

60  in.  -f-  12  =  5  ft. 

60  H-  12  =  5  ft. 

60  in.  H-  12  in.  =  5  ft. 

If  we  wish  to  reduce  60  in.  to  feet  we  have  three  correct  forms, 
any  one  of  which  is  easily  explained : 

60  X  Vi2  ft.  =  5  ft. 

60  in.  -r- 12  in.  =  5,  the  number  of  feet, 

60  -T-  12  =  5,  the  number  of  feet. 

If  slovenly  forms  are  allowed  here  they  must  be  expected  in  all 
subsequent  grades,  and  they  must  be  expected  to  lead  to  slovenly 
thought  in  the  treatment  of  all  kinds  of  problems. 

Review.  At  the  close  of  the  year  there  should  be  a  review  of 
all  the  essential  features  of  the  work  in  the  primary  grades. 
This  requires  skill  on  the  part  of  the  teacher  lest  it  become  stupid 
and  so  wearisome  as  to  lose  its  chief  value.  Original  local 


The  Work  of  the  Fourth  School  Year  101 

problems  to  test  the  children  in  the  four  fundamental  operations 
with  integers  and  (as  far  as  they  have  gone)  with  fractions, 
will  usually  render  the  work  interesting  and  will  hold  the 
attention. 

Nature  of  the  Problems.  Here  as  elsewhere  the  problems 
should  touch  the  children's  interests  and  be  adapted  to  their 
mental  abilities.  The  following  may  be  taken  as  types: 

Oral   Exercise 

1.  Tell  the  cost  of  some  kind  of  cloth.     How  much  will 
iol/2   yd.   cost? 

2.  Tell  the  cost  of  a  pair  of  shoes.    How  much  will  2  pairs 
cost? 

3.  If  a  man  earns  $3  for  10  hours'  work,  how  many  hours 
must  he  work  to  earn  enough  to  buy  his  daughter  a  pair  of 
shoes  at  $1.50? 

4.  How  many  hours  must  he  work  to  earn  enough  to  buy 
a  $6  suit  of  clothes  for  his  son? 

Written  Exercise 

1.  Sarah's  mother  bought  4Y5  yd.  of  cloth  for  a  cloak,  at 
$1.25  a  yard.     What  did  she  pay  for  it? 

2.  She  also  bought  3^  yd.  of  lining  at  SQC.  a  yard,  and 
4l/4  yd.  of  braid  at  2oc.  a  yard.    How  much  did  these  cost? 

3.  She  also  bought  6  pearl  buttons  at  $1.50  a  dozen,  and  2 
spools  of  silk  at  8c.  a  spool.     How  much  did  these  cost? 

4.  The  dressmaker  charged  $5  for  making  the  cloak.    What 
did  materials  and  making  cost? 

.5.  John's  mother  bought  2^  yd.  of  goods  for  a  coat,  at  $1.20 
a  yard,  and  2^  yd.  of  lining  at  48c.  a  yard.  How  much  did 
these  cost? 

6.  She  also  bought  a  dozen  buttons  at  25c.  a  dozen,  and  2 
spools  of  silk  at  8c.  a  spool,  and  paid  $3  for  making.  How 
much  did  the  coat  cost? 


CHAPTER  XIX 
THE  WORK  OF  THE  FIFTH  SCHOOL  YEAR 

The  Leading  Mathematical  Features.  There  should  in  this 
year  be  a  thorough  review  of  the  fundamental  operations  with 
integers.  This  should  be  followed  by  the  same  operations  with 
the  common  fractions  and  denominate  numbers  of  business. 
Percentage  may  be  begun,  although  in  some  places  it  is  better 
to  postpone  this  until  the  following  year. 

Review.  There  is  usually  a  new  text-book  begun  in  this 
grade,  and  this,  if  properly  arranged,  offers  plenty  of  material 
for  the  review  above  mentioned,  with  numbers  that  are  appro- 
priately larger.  Teachers  should  undertake  this  review  in  the 
spirit  and  for  the  reason  suggested  in  Chapter  XVI. 

Text-book.  The  new  text-book  begun  in  this  grade  will  natur- 
ally be  topical  in  its  arrangement,  that  is,  each  general  topic 
like  percentage  being  treated  once  for  all;  or  it  will  be  on  the 
plan  of  recurring  topics,  a  subject  like  percentage  being  met 
two  or  three  times.  As  has  already  been  said,  each  of  these 
types  has  its  advantages.  If  the  school  chooses  one  with  recur- 
ring topics  that  can  probably  be  followed  rather  closely.  If  on 
the  other  hand  it  adopts  one  arranged  by  topics  there  are  two 
courses  open :  ( i )  the  teacher  may  select  from  the  various  chap- 
ters such  material  as  fits  the  course  of  study  in  use  in  the  par- 
ticular locality,  a  task  of  no  great  difficulty;  (2)  the  book  may 
be  followed  closely,  the  pupils'  work  becoming  purely  topical. 
We  are  apt  to  condemn  the  latter  plan  because  it  is  old,  but 
perhaps  on  that  very  account  it  should  be  commended.  The 
world  has  used  it,  and  used  it  successfully,  and  it  has  the  merit 
that  it  brings  a  feeling  of  mastery,  a  sense  of  thoroughness,  and 
a  development  of  habit  that  is  sometimes  lacking  with  more 
modern  text-books.  In  general  it  may  be  said  to  depend  upon 
the  school  as  to  which  type  of  book  is  the  better,  and  as  to 
which  plan  of  using  the  topical  book  is  to  be  preferred.  In  a 

102 


The  Work  of  the  Fifth  School  Year  103 

school  system  with  a  reasonably  permanent  staff  of  teachers, 
with  adequate  supervision,  and  with  teachers'  meetings  that 
allow  classes  to  keep  in  touch  with  one  another,  the  book  with 
recurring  topics,  or  at  least  the  course  arranged  on  this  plan,  is 
undoubtedly  the  better.  It  is  more  psychological  and  it  allows 
for  a  better  grading  of  material.  On  the  other  hand  where 
teachers  change  frequently,  as  in  rural  schools,  it  is  safer  to  use 
the  topical  book  and  to  follow  it  rather  closely.  In  this  and 
the  following  chapters  the  arrangement  by  recurring  topics  is 
followed,  and  any  topical  text-book  can  easily  be  adapted  to  the 
sequence  suggested. 

Number  Space,  This  number  space  is  now  unlimited,  but 
names  beyond  billion  are  of  no  particular  importance.  Large 
numbers  should  always  represent  genuine  American  conditions. 
It  is  better  to  perform  several  operations  on  the  ordinary  num- 
bers of  daily  life  than  to  perform  one  on  an  absurdly  long 
number;  but  on  the  other  hand,  a  reasonable  number  of  opera- 
tions on  large  numbers  that  represent  real  business  cases  are 
to  be  commended. 

Addition.     Larger  numbers  and  longer  columns  may  now  be 
used,  but  there  is  a  limit  to  this  matter.   In  general,  the  numbers 
used  by  the  average  citizen  are  the  ones  to  drill  children  upon. 
Children  should  be  encouraged  to  read  columns  as  nearly  as  pos- 
sible as  they  read  a  word.     When  we  seek  the  word  "  book  " 
we  do  not  think  "  b,"  "  o,"  "  o,"  "  k,"— we  think  "book"  with- 
out any  spelling ;  so  when  we  see  the  annexed  column 
we  should  not  think,  "  6  and  3  are  9,  9  and  3  are          5 
12,  12  and  5  are  17,"  nor  even  "  6,  9,  12,  17,"  if  we          3 
can  do  better  than  this.     Probably  we  cannot  train           3 
our  eyes  to  see  17  at  a  glance,  as  we  seek  "  book,"          6 
but  it  is  well  to  encourage  children  to  look  at  this 
as  9  +  8,  thinking  of  the  6  and  3  as  9,  and  the  3 
and  5  as  8.    But,  however  we  think  of  such  a  column,  we  should 
always  check  our  result  by  adding  in  the  reverse  order.     If 
teachers  do  not  think  this  necessary,  let  them  add  twenty  sets 
of  say  ten  five-figure  numbers  each,  working  rapidly,  and  see 
how  many  mistakes  they  themselves  will  make. 

Subtraction.  This  subject  has  been  sufficiently  discussed  on 
page  45.  The  important  matter  is  not  now  the  explanation, 


IO4  The  Teaching  of  Arithmetic 

for  the  technique  has  already  been  learned;  the  operation,  ac- 
curately and  rapidly  performed,  is  the  desideratum,  the  check 
being  of  great  importance  in  securing  the  essential  accuracy. 

Multiplication.  It  is  now  advisable  to  let  the  children  know 
some  good,  practical  check  on  their  work  in  multiplication,  such 
as  computers  actually  use.  Of  the  checks,  the  simplest  is  that 
of  "  casting  out  9's."1 

Division.  The  children  are  now  old  enough  to  understand 
the  two  forms  of  division  illustrated  by  the  following: 

$125.  ^-$5  =  2$ 
$125  -4-  25  =  $5 

There  are  no  generally  accepted  names  to  distinguish  these, 
"  measuring  "  and  "  partition  "  not  meaning  much  to  children. 
It  suffices  that  it  is  clear  that  there  are  these  two  forms,  and  to 
see  that  we  avoid  such  inaccuracies  as  $125  -r-  $25  =  5  cows. 

Factors  and  Multiples.  This  subject  formerly  played  a.  very 
important  part  in  arithmetic,  when  large  fractions  had  to  be 
reduced  to  lower  terms.  With  the  introduction  of  the  decimal 
fraction  about  1600,  however,  it  lost  much  of  its  former  im- 
portance and  need  play  but  a  small  part  in  the  arithmetic  of 
to-day.2 

Common  Fractions.  Some  objective  work  will  still  be  neces- 
sary in  treating  common  fractions  but  it  should  be  dispensed 
with  as  soon  as  possible  and  the  material  should  not  be  of  one 
kind  alone.  In  the  operations  children  should  not  be  required 
to  give  very  elaborate  explanations,  although  they  should  see 
clearly  the  reasons  at  the  time  they  learn  the  processes.  This 
has  been  discussed  already  in  Chapter  IX  and  may,  therefore, 
be  dismissed  at  this  time. 

Denominate  Numbers.  The  operations  with  these  numbers 
should  be  a  part  of  the  work  of  the  year,  but  only  practical 
cases  should  be  taken.  To  divide  a  compound  number  of  four 


1  The  explanation  of  this  process   is  too  long  to  be  given  here.     The 
reader  may  consult  the  author's  Handbook,  p.   57,   Beman   and    Smith's 
Higher  Arithmetic,  or  the   appendix  to  the  Wentworth- Smith  Complete 
Arithmetic  and  the  Arithmetic,  Book  III. 

2  For  a  theoretical  treatment  of  the  subject  from  the  advanced  stand- 
point, consult  Beman  and  Smith's  Higher  Arithmetic. 


The  Work  of  the  Fifth  School  Year  105 

denominations  by  another  one  of  three,  for  example,  consumes 
time  and  patience  to  no  worthy  purpose. 

How  to  Solve  Problems.  Inasmuch  as  the  children  now  begin 
to  consider  problems  of  more  than  two  steps,  it  becomes  neces- 
sary to  devote  more  attention  to  the  methods  of  solving  ex- 
amples. The  step  form  of  analysis,  therefore,  has  a  legitimate 
place  in  this  year's  work.  If  teachers  hope  for  exactness  of 
thought  they  must  insist  upon  accuracy  of  statement  in  these 
written  exercises. 

Percentage  and  Decimals.  The  study  of  decimal  fractions 
may  safely  be  undertaken  in  this  grade,  and  this  may  be  fol- 
lowed, if  desired,  by  an  elementary  treatment  of  percentage.  If 
at  the  outset  children  understand  that  6%  is  only  another  way 
of  writing  yf^  and  0.06,  there  will  be  but  little  difficulty  in 
introducing  percentage.  One  important  feature  is  the  inter- 
change of  the  per  cent  forms,  decimal  fractions,  and  common 
fractions,  as  for  example,  in  ]/$  =-r^=  0.25  =  2$%.  It  is 
better  not  to  introduce  any  formulas  or  rules  in  such  work  in 
percentage  as  may  be  taken  at  this  time,  but  to  analyze  each 
problem  as  it  arises.  In  the  next  school  year  it  is  allowable 
to  reverse  this  policy. 

Discount.  Of  all  the  applications  of  percentage  the  most 
common  is  'discount,  and  it  is  at  the  same  time  the  simplest. 
This  topic  may,  therefore,  be  introduced  in  this  grade,  the  other 
applications  being  reserved  for  the  sixth  year. 

Nature  of  the  Problems.  The  great  industries  of  the  country 
may  be  taken  up  at  this  time  as  a  profitable  field  for  the  applica- 
tions of  arithmetic.  Children  now  begin  to  know  enough  geogra- 
phy to  permit  of  this  wider  view,  and  problems  that  relate  to 
their  own  country  have  an  interest  that  the  traditional  ones  about 
the  man  who  "  owned  a  field  of  corn  "  lacked.  Such  problems 
are  not  statistical  to  the  extent  that  their  data  are  to  be  memor- 
ized, but  they  state  real  conditions  instead  of  false  ones.  Of 
problems  suited  to  this  grade  the  following  are  types  relating 
to  the  production  of  corn,  one  of  the  great  food  products  of  the 
country.  The  greatest  corn-producing  states  are  Iowa,  Illinois, 
Nebraska,  Missouri,  Kansas,  and  Indiana,  and  such  work  may 
be  taken  in  connection  with  the  study  of  the  geography  of  these 


io6  The  Teaching  of  Arithmetic 

sections  whenever  this  can  be  brought  about  without  too  great 
change  in  the  curriculum. 

1.  When  this  country  produced  2,105,102,400  bu.  of  corn  a 
year,   averaging  25  bu.  to  the  acre,  how  many  acres  had  we 
in  corn? 

2.  If  3  bu.  of  corn  could  then  be  bought  for  $i,  what  was 
the  total  value  of  this  yield  of  2,105,102.400  bu? 

3.  When  Iowa's  annual  product  amounted  to  305,800,000  bu., 
this  was  how  many  times  the  440,000  bu.  produced  by  Maine? 

4.  To  transport   1000  Ib.  of  corn  from  St.  Louis  to  New 
Orleans  by  river  costs  $i.     How  much  will  it  cost  to  transport 
1750  tons? 

5.  If  the  average  value  of  corn  for  each  of  the  46,610  acres 
given  to  it  in  Connecticut  in  a  certain  year  was  $21,  and  for  each 
of  the  4,031,600  acres  in  Indiana  $13,  what  was  the  entire  value 
of  the  corn  crop  of  each  state? 

6.  If  the  average  annual  corn  crop  per  acre  is  40  bu.  in 
Wisconsin,  36  bu.  in  Maine,  37  bu.  in  New  Hampshire,  38  bu. 
in  Massachusetts,  38  bu.  in  Indiana,  and  38  bu.  in  Iowa,  find 
the  average  by  adding  and  dividing  by  6. 


CHAPTER  XX 
THE  WORK  OF  THE  SIXTH  SCHOOL  YEAR 

The  Leading  Mathematical  Features.  The  leading  features 
of  this  year  should  be  percentage  and  its  applications,  particu- 
larly to  discount,  profit  and  loss,  commission,  and  interest. 
Ratio  and  simple  proportion  may  also  be  included. 

The  General  Solution  of  Problems.  Since  the  work  in  per- 
centage introduces  the  pupil  to  the  problems  of  business,  some  of 
which  become  rather  intricate  in  the  later  school  years,  it  is  well 
at  this  time  to  take  up  rather  systematically  questions  of  the  solu- 
tion of  problems  in  arithmetic.  To  this  end  there  should  be  con- 
sidered exercise  in  analysis  in  general  and  in  unitary  analysis  in 
particular,  and  the  equation  may  well  begin  to  find  place  in  the 
mental  equipment  of*jjae*  child.  As  to  the  matter  of  analysis  no 
question  will  be  raised,  but  as  to  introducing  the  letter  x  some 
teachers  are  In  doubt.  When,  however,  we  come  to  consider  that 
it  merely  replaces- an  awkward  symbol  that  has  long  been  used, 
and  makes  the  work  much  clearer,  the  objection  cannot  be  main- 
tained. For  example,  2  +  (  ?)  =  7  is  sometimes  used  as  early 
as  the  first  schodl  "y.ear  ^.ttfis^ho wever,  is  only  a  complicated  way 
of  writing  2  -K$-=  7,  tr^e  two  meaning  exactly  the  same  thing. 
Similarly,  4  :  7  =M2*:  (?)  is  only  an  awkward  way  of  writing 
what  is  equivalent*?© 

x        7 


12  \    4 

the  latter  being  in  every  way  simpler  of  understanding  and 
easier  of  solution.  Theje  are  several  classes  of  problem  in  per- 
centage that  are  made  clearer' by  the  use  of  this  convenient  x, 
and  its  use  is  quite  as  arithmetical  as  algebraic. 

Percentage.     In  this  work  special  attention  should  be  given 
to  the  common  per  cents  and  fractions  of  business,  such    as 

and   so   on. 
107 


io8  The  Teaching  of  Arithmetic 

In  the  matter  of  solution,  the  x  should  be  used  in  those  inverse 
cases  where  it  makes  the  problem  clearer.  Such  is  the  case  of 
finding  the  cost  of  goods  that  sell  at  10%  above  cost,  and  sell  for 
$126.50.  Here  we  have,  if  x  represents  the  cost, 

x  +  .\QX  =  $126.50 
1. 10^  =  $126. 50 

X  =  $126.50  -4-  1. 10 
#==$115 

Other  forms  of  solution  might  be  used,  but  this  is  the  most 
satisfactory. 

Discount.  This  being  the  first  and  most  important  of  the 
applications  of  percentage,  considerable  attention  should  be  de- 
voted to  it.  The  case  of  several  discounts  may,  however,  be 
postponed  until  the  following  year. 

Profit  and  Loss  on  Purchases.  This  topic,  so  closely  con- 
nected with  the  business  world  with  which  the  child  is  now 
coming  into  closer  contact,  may  claim  to  rank  second  in  impor- 
tance among  the  applications  of  percentage.  The  principles 
involved  are  very  simple,  particularly  if  one  allows  the  letter  x 
to  throw  light  upon  all  inverse  problems.  The  examples  should 
follow  as  closely  as  possible  the  common  business  customs  of 
the  mercantile  world. 

Commission.  This  topic  ranks  possibly  third  in  importance 
among  the  applications  of  percentage.  A  considerable  field  of 
applications  exists,  particularly  in  relation  to  the  sending  of  farm 
produce  to  the  cities.  The  problems  can,  therefore,  be  made  to 
seem  real  to  the  children,  whether  they  live  in  the  country  or 
see  farm  products  for  sale  in  the  city. 

Interest.  This  subject  may  already  have  been  met  by  the 
children.  It  is  now  taken  up  and  extended  to  more  difficult 
questions.  Only  real  cases  should,  however,  be  considered.  For 
example,  in  this  school  year,  at  least,  there  is  little  advantage  in 
trying  to  find  the  capital,  given  the  rate,  time,  and  interest.  It  is 
better  to  spend  time  in  writing  promissory  notes  and  in  comput- 
ing the  interest,  than  to  put  it  on  questions  that  seldom  arise  in 
business  life.  If  we  wish  more  complicated  problems  they  are 
easily  secured  from  genuine  mercantile  sources. 

Ratio.  This  may  be  introduced  this  year  or  reserved  for  the 
seventh  grade.  It  was  formerly  introduced  merely  as  an  intro- 


The  Work  of  the  Sixth  School  Year  109 

duction  to  proportion.  It  is  easy,  however,  to  see  that  it  may  be 
of  some  i^se  by  itself,  and  teachers  are  advised  to  consider  this 
phase  of  tfce  subject.  We  mix  fertilizers  on  the  farm  in  a  given 
ratio,  we  find  ratios  of  attendance  to  absence  in  the  school,  and 
the  term  is  used  in  the  same  way  in  business  life. 

Proportion.  This  subject  may  also  be  delayed  another  year. 
It  has  lost  a  good  deal  of  its  importance  of  late.  A  proportion 
is,  as  shown  on  page  107,  merely  one  method  of  writing  a  simple 
equation,  and  with  the  letter  x  allowed  in  school,  the  equation 
form  is  likely  to  replace  that  of  proportion.  When  this  is  not 
the  case,  ordinary  analysis  is  likely  to  be  substituted  for  pro- 
portion. For  example,  consider  this  problem:  If  a  shrub  4  ft. 
high  casts  a  shadow  6  ft.  long  at  a  time  that  a  tree  casts  one 
54  ft.  long,  how  high  is  the  tree? 

Here  we  may  write  a  proportion  in  the  form,  6  ft.  :  4  ft.  = 
54  ft.:  (?),  not  attempting  to  explain  it,  but  applying  only  an 
arbitrary  rule.  This  is  the  old  plan.  Or  we  may  put  the  work 
into  equation  form. 


54       6 

and  deduce  the  rule  for  dividing  the  product  of  the  means  by  the 
given  exteme.  Or  we  may  take  the  same  equation  and  get  our 
result  easily  by  multiplying  these  equals  by  54,  giving 

*  =  36 

Or  we  may  say :  If  a  6  ft.  shadow  is  cast  by  a  4  ft.  object,  a  I  ft. 
shadow  would  be  cast  by  a  */e  **•  object,  and  a  54  ft.  shadow 
would  be  cast  by  a  54  X  4/e  ft.  object,  or  a  36  ft.  object. 

Of  these  plans  the  first  is  the  most  difficult  to  explain;  the 
rest  are  equally  easy,  and  the  third  is  the  shortest. 

Measures.  The  work  in  measures  this  year  may  be  confined 
to  simple  surfaces  and  solids,  and  may  properly  include  practical 
cases  of  house  building,  plastering,  carpentering,  and  the  like. 
Here  is  a  real  field,  interesting  and  profitable.  Proportion  leads 
to  exercises  in  similar  figures,  and  this  has  some  excellent  appli- 
cations in  lumbering  and  in  carpenter's  work. 

Nature  of  the  Problems.  In  each  succeeding  year  the  prob- 
lems now  come  to  relate  more  and  more  to  the  industries  of  the 


no  The  Teaching  of  Arithmetic 

people,  and  the  range  of  applications  becomes  very  great.  The 
farm  child  learns  not  only  of  his  own  surroundings  but  of  the 
great  industries  of  the  city,  while  to  the  city  child  the  great  story 
of  the  soil  and  its  products  opens  up  a  new  world.  The  follow- 
ing farm  problems  may  be  taken  as  types  of  the  problems  suited 
to  this  grade: 

1.  A  farmer  puts  5  acres  into  celery,  setting  out  20,000  plants 
to  the  acre.    The  yield  being  1,500  doz.  heads  to  the  acre,  what 
is  the  ratio  of  the  plants  matured  to  the  others?  '•    ... 

2.  He  pays  $95  an  acre  for  seeds,  fertilizers,  labor,  and  other 
expenses,  and  sells  the  crop  at  I5c.  a  dozen  heads.    What  is  his 
profit  on  the  5  acres  ?j?    fciTO.oo 

3.  Another  farmer  tries  setting  out  30,000  plants  to  the  acre, 
but  only  80%  mature,  and  these  are  so  small  that  he  has  to  put 
16  in  a  bunch  to  sell  for  a  dozen,  and  then  gets  only  i4c.  a 
bunch.     His  expenses  are  $100  an  acre.     At  this  rate  what  is 
his  profit  on  5  acres  ?£f  JTlTO.O* 

4.  A  farmer  has  a  3O-acre  meadow  yielding  il/2  tons  of  hay 
to  the  acre.     If  by  spending  $300  a  year  for  fertilizers,  he  can 
bring  the  yield  to  4  tons  to  the  acre,  how  much  more  will  he  make 
a  year,  hay  being  worth  $8  a  ton? 

5.  A   farmer   reads   that   a  good   mixture  of   seed   for  his 
meadow  is,  by  weight,  as  follows:  timothy  40%,  redtop  40%, 
red  clover  making  up  the  rest.    At  40  Ib.  of  seed  to  "the  acre, 
how  many  pounds  of  each  should  he  sow?      t  to  "T-  I  r>  Jf--    t 

6.  The  following  is,  by  weight,  a  good  mixture  of  seed  for  a 
~J        pasture:  Kentucky  blue  grass$25%,  white  clover  \2,y2%;  per- 

ennial rye  28%%,  red  fescue  $£%,  redtop  25%.  At  32  Ib.  to 
the  acre,  how  many  pounds  of  each  are  used? 

7.  A  cow  weighing  1000  Ib.  consumes  the  equivalent  of  3>4 
tons  (2000  Ib.  to  the  ton)  of  dry  fodder  a  year;  a  loo-lb.  sheep, 
770  Ib.  ;  every  ton  of  live  pork,  12  tons  ;  and  every  ton  of  live 
horseflesh,  8.4  tons.     Each  class  of  animals  consumes  what  per 
cent  of  its  own  weight  of  drv  fodder  a  year? 


T  ^ 


CHAPTER  XXI 
THE  WORK  OF  THE  SEVENTH  SCHOOL  YEAR 

The  Leading  Mathematical  Features.  As  in  the  preceding 
grade,  it  is  well  to  begin  by  a  general  review  of  the  fundamental 
processes  from  a  higher  standpoint  than  before.  Ratio  and  pro- 
portion are  usually  completed  in  this  year,  whether  introduced 
here  for  the  first  time  or  not,  and  the  applications  naturally  cover 
a  broader  field.  Percentage  is  the  leading  topic  of  the  year. 

Our  Numbers.  The  children  are  now  ready  to  consider  the 
writing  of  numbers  from  a  higher  standpoint,  to  know  something 
of  the  interesting  history  of  the  numerals  they  use  and  of  the 
science  of  arithmetic  that  they  are  studying.  The  story  of  the 
Roman  numerals,1  and  that  of  the  Arabic  numerals  make  these 
subjects  seem  more  real  at  this  time.  The  difference  between 
a  uniform  scale,  as  seen  in  our  system  of  money,  and  a  varying 
one,  as  seen  in  the  English  system,  should  also  be  explained, 
and  the  advantage  of  the  former  understood.  The  relation  be- 
tween integers,  the  various  kinds  of  fractions,  and  compound 
numbers,  may  now  understandingly  be  taken  up. 

The  Fundamental  Operations.  These  may  now  be  reviewed 
in  such  way,  that  is  by  the  introduction  of  such  new  material, 
as  to  maintain  the  interest  even  in  an  old  subject.  This  is  par- 
ticularly true  if  the  teacher  will  now  and  then  suggest  such  short 
methods  as  may  be  found  in  most  of  the  advanced  arithmetics. 
The  check  of  casting  out  nines  should  now  be  used  for  all 
products  and  quotients.  It  is  simple,  it  takes  but  a  moment, 
and  it  checks  most  of  the  errors  that  are  liable  to  arise.  It 
cannot  be  too  much  impressed  upon  teachers  and  pupils  that 
both  are  very  liable  to  errors  in  all  kinds  of  calculation,  and 
that  they,  like  business  computers,  should  always  apply  some 
kind  of  check  to  every  result  obtained.  Some  teachers  feel  that 


1  This  is  told  in  condensed  form  in  the  author's  Handbook  to  Arithmetic. 
See  also  the  Bibliography  at  the  close  of  Chapter  I. 

ill 


112  The  Teaching  of  Arithmetic 

the  work  should  be  so  accurately  done  that  checks  should  be 
unnecessary.  This  is  a  good  theory,  but  practically  it  will  not 
work  even  with  the  ones  who  advocate  it.  No  good  professional 
computer  would  think  of  leaving  his  results  without  checking 
them,  and  if  a  professional  will  not  do  this,  why  should  we 
expect  a  child  to  be  so  infallible  as  to  do  it? 

Measures.  All  tables  of  measure  in  common  use  should  be 
reviewed  in  this  year.  If,  in  this  review,  some  historical  notes 
are  given  on  the  origin  of  such  measures  as  the  yard,  inch,  foot, 
mile,  quart,  gallon,  and  acre,  the  pupils  will  find  the  work  taking 
on  a  new  interest.  Teachers  are  advised  that  it  is  of  little  value 
to  memorize  facts  that  will  not  be  used  in  practical  life.  If  we 
wish  to  know  the  number  of  cubic  inches  in  a  bushel  we  may  go 
to  an  encyclopedia  or  a  dictionary;  it  is  surdy  inadvisable  to 
burden  our  minds  with  such  details. 

Longitude  and  Time.  This  subject  has  greatly  changed  within 
a  few  years.  To-day  most  of  the  civilized  world  uses  some 
form  of  standard  time.  Therefore,  our  attention  may  properly 
be  confined  to  the  geographical  principle  involved,  to  the  prob- 
lem of  standard  time,  and  to  the  question  of  longitude  at  sea. 
Teachers  are  urged  not  to  allow  slovenly  work  in  this  subject 
under  the  plea  that  bad  forms  bring  true  results  in  a  shorter 
time  than  good  forms.  This  matter  has  been  sufficiently  dis- 
cussed on  page  30,  and  the  chapter  on  longitude  and  time  seems 
to  be  one  of  the  worst  offenders  in  all  arithmetic.  A  form 
like  45  -r- 15  =  3  hrs.  is  false  and  serves  to  undo  all  of  the  good 
to  be  derived  from  the  topic. 

Percentage.  This  topic,  so  vital  in  business  life  to-day,  should 
be  touched  upon  several  times  in  the  elementary  school.  If 
the  work  is  sufficiently  progressive  the  pupils  will  not  find  that 
"  the  edge  of  interest "  is  worn  off.  In  this  year  there  should 
be  a  good  deal  of  oral  work  in  the  common  per  cents  of 
business,  pupils  coming  to  feel  that  pencil  and  paper  are  unneces- 
sary in  finding  12%%,  25%,  3^/3%,  50%,  662/3%,  and  75%  of 
ordinary  numbers.  As  to  the  use  of  terms  like  "  base,"  "  rate," 
"  percentage,"  "  amount,"  and  "  difference,"  there  is  little  that  can 
be  said  in  their  favor.  They  were  invented  in  the  rule  stage  of 
arithmetic,  and  have  served  their  purpose.  Of  course,  we  need 
"  rate,"  it  being  a  stock  term  of  the  business  world.  "  Percent- 


The  Work  of  the  Seventh  School  Year  113 

age  "  is,  however,  rather  confusing  than  otherwise,  ( i )  because 
it  is  understood  by  the  pupils  as  the  name  of  the  subject  as  a 
whole,  and  (2)  because  the  business  world  does  not  use  it  quite 
as  the  school  does.  "  Base  "  means  so  many  things  in  mathe- 
matics that  its  use  is  equally  confusing,  while  of  "  amount "  and 
"  difference "  this  is  still  more  noticeably  the  case.  On  the 
whole,  therefore,  it  is  as  well  not  to  use  these  terms,  although 
they  are  found  in  most  of  our  leading  books  to-day  because  of 
the  demands  of  teachers. 

It  should  also  be  remarked  that,  if  the  use  of  x  is  allowed, 
there  is  no  excuse  for  the  old  formulas  of  percentage.  They 
are  nothing  but  condensed  rules ;  if  they  are  not  explained  they 
defeat  part  of  the  purpose  of  studying  arithmetic;  if  they  are 
explained  they  are  much  harder  than  the  equation  form  with 
the  single  letter  x. 

It  is  well  to  bear  constantly  in  mind,  in  the  midst  of  the  large 
number  of  possible  cases  of  percentage,  that  the  two  important 
things  in  the  subject  are  these:  (i)  to  find  some  per  cent  of  a 
given  number,  and  (2)  to  find  what  per  cent  one  number  is  of 
another.  All  the  rest  is  relatively  unimportant,  and  on  these 
two  the  emphasis  should  accordingly  be  laid. 

Simple  Interest.  This  is  the  leading  application  of  percentage 
in  this  year,  and  the  attention  of  pupils  should  be  concentrated 
on  the  single  problem  of  finding  interest  in  practical  cases.  To 
find  the  time,  given  the  principal,  rate,  and  interest,  is  of  very 
slight  importance,  and  so  for  other  similar  cases ;  but  to  find  the 
interest,  that  is  the  great  point. 

Ratio  and  Proportion.  This  work  should,  as  stated  in  the 
preceding  grade,  be  confined  largely  to  the  treatment  of  practical 
questions,  and  there  are  only  a  few  where  this  subject  can  be 
used  to  real  advantage.  These  are  chiefly  related  to  similar 
figures,  although  some  other  questions,  like  those  of  simple 
physics,  enter.  Compound  proportion  has  little  reason  to  claim 
a  place  in  our  schools  to-day.  If  explained,  the  process  is  a 
very  hard  one;  if  not,  it  is  a  useless  one,  since  we  now  have 
better  methods  of  solving  problems. 

Nature  of  the  Problems.  With  each  succeeding  school  year 
the  children  develop  new  interests  and  come  nearer  to  the  great 
world  that  they  are  soon  to  enter.  The  range  of  topics  is  now 


114  The  Teaching  of  Arithmetic 

practically  unlimited,  and  the  opportunities  for  offering  series  of 
related  problems  are  excellent.  As  a  type  of  such  problems  the 
following  may  be  given,  appealing  this  time  to  the  girls,  who 
are  usually  rather  neglected  in  the  matter  of  applied  arithmetic : 

Dressmaking  Problems 
Written  Exercise 

1.  A  dressmaker  bought  16  yd.  of  velvet  at  $3  a  yard,  selling 
9  yd.  at  a  profit  of  i62/^%  and  the  rest  at  a  rate  of  profit  half 
as  great.     What  was  the  rate  of  gain  on  the  whole? 

2.  She  bought  a  25~yd.  box  of  chiffon  velvet  at  $4  a  yard, 
with  10%  off  for  cash,  selling  it  at  $4.35  a  yard.    What  was  her 
gain  per  cent? 

3.  She  bought  a  75-yd.  piece  of  silk  skirt  lining  at  65c.  a 
yard.    She  sold  28  yd.  at  9oc.,  15  yd.  at  95c.,  and  the  remainder, 
at  the  close  of  the  season,  at  /oc.    What  was  her  per  cent  of  gain  ? 

4.  She  bought  a  5o-yd.  piece  of  silk  waist  lining  at  75c.  a 
yard.    She  sold  12  yd.  at  $i  and  10  yd.  at  95c.,  but  the  remainder, 
being  kept  in  stock  over  the  season,  had  to  be  sold  at  65c.    What 
was  her  per  cent  of  gain  or  loss? 

5.  She  bought  a  2O-yd.  silk  dress  pattern  at  $2.10  a  yard, 
being  allowed,  as  a  dressmaker,  a  discount  of  $%,  and  6%  off 
for  cash.     She  charged  her  customer  the  marked  price,  $2.10. 
What  was  her  per  cent  of  profit? 

6.  She  charged  her  customer  $25.50  for  3  yd.  of  Honiton 
lace,  which  had  cost  her  $7  a  yard.    What  was  her  per  cent  of 
profit  ? 

7.  She  charged  her  customer  $2  for  findings  for  the  dress. 
These  consisted  of  4  spools  of  silk  at  IDC.  each,  i  spool  of  thread 
at  5c->  3  yd-  of  featherbone  at  ioc.,  a  card  of  hooks  and  eyes 
at  8c.,  skirt  braid   i6c.,  plaiting  3oc.,  waist  binding  3Oc.,  and 
collar  ioc.     What  was  her  gain  per  cent  on  the  findings? 


CHAPTER  XXII 


The  Leading  Mathematical  Features.  The  work  this  year  is  in 
the  line  of  business  applications,  including  advanced  mensuration. 

Business  Applications.  The  boy  and  girl  should  now  begin 
to  feel  that  the  world  of  business  and  of  life  is  opening  before 
them.  It  should  therefore  be  the  duty  of  the  school,  even  more 
than  in  the  preceding  grades,  to  apply  arithmetic  to  the  genuine 
problems  of  life,  particularly  with  reference  to  the  common  occu- 
pations of  the  people. 

Banking.  In  banking,  for  example,  we  should  not  seek  to 
train  accountants  or  bookkeepers  or  cashiers,  but  we  should  seek 
to  give  a  fair  idea  of  the  duties  of  these  men  in  the  ordinary 
savings  bank  and  bank  of  deposit.  A  girl,  for  example,  needs 
to  know  how  to  deposit  money  in  a  bank  and  how  to  draw  checks 
as  well  as  a  boy,  and  such  operations  should  become  as  real 
as  the  school  can  make  them.  School  banks,  with  deposit  slips, 
checks,  bank  book,  cashier,  paying  teller,  and  receiving  teller, 
should  assist  in  this  work. 

Partial  Payments.  This  subject  has  not  the  practical  value 
that  it  had  when  banks  were  not  so  numerous  as  now,  and  when 
their  machinery  was  not  perfected.  The  old-style  problem  in 
partial  payments  should  therefore  give  place  to  the  more  prac- 
tical cases  found  in  our  best  modern  books. 

Partnership.  This  is  another  subject  that  has  entirely  changed 
within  a  short  time.  The  stock  company  (corporation)  has 
largely  supplanted  it,  save  in  its  simplest  form.  The  work  of 
the  schools  should  therefore  be  confined  to  this  common  form, 
the  obsolete  ones  being  supplanted  by  work  on  corporations. 

Simple  Accounts.  It  is  not  worth  while  to  teach  an  elaborate 
form  of  bookkeeping  to  the  average  citizen.  On  the  other  hand 
it  is  necessary  that  every  one  should  know  how  to  keep  simple 
accounts,  and  this  work  should  be  taken  up  in  this  year.  It 
should  relate  to  the  income  and  expenditures  of  daily  life,  in  the 

"5 


u6  The  Teaching  of  Arithmetic 

home,  on  the  farm,  or  in  the  shop,  rather  than  to  the  technical 
needs  of  the  merchant,  the  latter  being  part  of  the  special  train- 
ing of  the  individual  who  enters  this  line  of  trade. 

Exchange.  Here  again  there  has  been  a  great  change  within 
a  few  years.  The  form  of  time  draft  given  in  most  of  the  old- 
style  arithmetics  has  given  place  either  to  sight  drafts  or  to 
another  kind  of  time  draft.  Teachers  should  therefore  be 
particular  to  use  only  those  types  that  the  ordinary  citizen  meets 
to-day,  about  which  girls  and  boys  alike  should  be  informed.  In 
connection  with  this  work  a  short  talk  upon  the  clearing  house, 
upon  which  any  bank  will  gladly  inform  the  teacher,  will  add 
new  interest. 

The  Metric  System.  This  system  might  be  taught  much 
earlier  than  the  eighth  school  year,  and  there  would  be  some 
advantage  in  so  doing.  But  when  we  consider  that  it  is  not  yet 
used  practically  by  many  Americans,  it  seems  as  well  to  postpone 
it  until  this  time.  There  are  three  chief  reasons  for  teaching 
it  now:  (i)  General  information  requires  us  to  know  a  system 
that  is  used  by  a  large  part  of  the  civilized  world,  excluding  the 
English-speaking  portion;  (2)  it  is  used  in  all  scientific  labor- 
atories in  America;  (3)  our  people  should  be  sympathetic  with 
a  system  that  is  liable  to  replace  our  own  before  long  in  all 
matters  relating  to  our  growing  foreign  trade;  if  we  sell  ma- 
chines abroad,  the  measurements  must  be  metric  in  most  cases, 
and  to  foster  this  trade  many  of  our  skilled  workmen  will 
eventually  need  to  use  these  instead  of  the  awkward  ones  with 
which  we  are  familiar. 

At  the  same  time  we  must  not  go  to  an  absurd  extreme,  but 
must  remember  that  our  common  system  is  the  one  that  the 
people  use  and  that  the  children  must  know  before  all  others. 
In  teaching  the  metric  system  the  results  will  be  poor  unless  the 
children  use  the  actual  measures  and  come  to  visualize  the  basal 
units  as  they  should  in  their  own  system. 

Taxes.  This  topic,  like  others  of  practical  life,  should  be 
treated  from  the  standpoint  of  local  conditions  as  far  as  possible. 
It  should  include  the  question  of  tariff,  and  a  few  brief  talks  on 
civics  should  make  the  whole  question  a  real  one  for  the 
pupils. 

Insurance.  This  subject  has  become  so  technical  that  all  that 
the  schools  can  hope  to  do  is  to  give  a  general  conception  of  the 


The  Work  of  the  Eighth  School  Year  117 

work  of  the  various  kinds  of  companies,  and  to  confine  the  prob- 
lems to  the  simplest  practical  cases  that  the  people  need  to  know 
about.  We  should  not  attempt  to  enter  upon  the  technicalities 
of  agent  work,  nor  to  do  more  than  explain  briefly  some  of  the 
common  types  of  policy. 

Corporations.  As  remarked  under  Partnership,  the  corpo- 
ration has,  for  good  or  evil,  replaced  the  individual  in  large 
business  ventures.  Our  schools  must,  therefore,  adjust  their 
work  to  this  change.  Pupils  should  know  what  a  corporation 
is,  its  chief  officials,  how  it  is  legally  organized,  what  stocks  and 
bonds  are,  how  dividends  are  declared  and  paid,  and  the  legiti- 
mate work  of  stock  exchanges.  On  the  other  hand  the  schools 
cannot  be  expected  to  teach  the  technicalities  of  the  stock  brok- 
er's office,  nor  to  supply  information  beyond  that  needed  by  the 
general  citizen.  The  newspaper  stock  reports  furnish  an  excel- 
lent basis  for  the  practical  problems  that  the  case  demands. 

Powers  and  Roots.  For  purposes  of  mensuration  square  root 
is  necessary.  Cube  root  may  well  be  delayed  until  the  pupil 
studies  algebra,  because  it  has  so  few  practical  applications. 
Even  square  root  is  more  valuable  as  a  bit  of  logic  than  as  a 
practical  subject,  since  those  who  use  it  most  employ  tables. 
The  explanation,  therefore,  is  even  more  important  than  the 
technique  of  the  work,  and  children  of  this  age  can  easily  com- 
prehend it,  either  by  the  use  of  the  diagram  or  by  the  formula, 
the  latter  being  quite  easily  understood  by  this  time. 

Mensuration.  This  work  is  now  completed  so  far  as  the 
needs  of  the  average  person  are  concerned.  The  teacher  should 
use  simple  models  that  can  be  made  in  the  school  room,  as  sug- 
gested in  the  best  arithmetics.  It  is  not  expected  that  strict 
geometric  demonstrations  can  be  given,  but  it  is  entirely  possible 
to  avoid  arbitrary  rules  by  giving  enough  objective  work  to  make 
the  matter  clear.  It  is  not  advisable  to  introduce  work  that  is 
not  used  in  ordinary  life,  such  as  finding  the  volume  of  a  frustum 
of  a  cone,  there  being  a  sufficient  amount  of  more  important 
work  to  occupy  the  time  and  attention  of  pupils. 

Nature  of  the  Problems.  The  problems  should  appeal  to  the 
business  needs  that  are  soon  to  come  to  the  children,  and  the 
following  are  suggested  as  types: 

I.  A  boy  who  has  been  working  this  year  at  $25  a  month  is 
offered  either  an  increase  of  20%  for  next  year  or  a  salary  of  $7 


n8  The  Teaching  of  Arithmetic 

*• 

a  week.     Which  will  bring  the  more  income,  and  how  much 
more  per  year?     (Use  52  wk.) 

2.  A  girl  who  has  been  working  in  a  factory  at  $21.67  a 
month  is  offered  an  increase  of  10%  where  she  is  or  a  salary  of 
$5.60  per  week  elsewhere.    Which  will  bring  the  more  income, 
and  how  much  more  per  year?     (Use  52  wk.) 

3.  A  boy  went  to  work  at  9<Dc.  a  day.    The  second  year  his 
wages  were  increased  20%,  the  third  year  they  were  42c.  a  day 
more  than  the  second,  and  the  fourth  they  were  increased  ZZY^%- 
At  300  working  days  to  the  year,  what  was  his  total  income  for 
each  year? 

4.  A  girl  entering  a  trade  school  finds  that  graduates  from 
the  dressmaking  department  receive  on  an  average  $4.60  a  week 
the  first  year;  those  from  the  millinery  department,  5%  less; 
those  from  the  embroidery  department,  5%  more  than  the  dress- 
makers;  and  those  from  the  operating  department  66^3%   as 
much  as  the  last  two  together.    Find  the  average  wages  of  each, 
and  tell  which  department  the  girl  probably  entered.  (Use  52  wk.) 

5.  A  girl  leaving  the  public  school  finds  she  can  enter  a  city 
shop  at  a  salary  of  $3  a  week  the  first  year,  with  16%%  more 
the  second  year,  and  a  14* /7%  increase  the  third  year.     Instead 
of  this  she  enters  a  trade  school  for  a  year,  tuition  free.     She 
then  receives  a  salary  of  $5  a  week  the  first  year  and  20%  more 
the  second  year.    Counting  50  working  weeks  a  year,  how  much 
more  does  she  receive  in  three  years  by  the  plan  she  follows 
after  leaving  the  public  school  than  she  would  have  received 
without  the  trade-school  training? 

A  Comparison  of  Eighth  Grade  Work.  It  is  well  known 
that  in  Europe  the  specialization  of  schools  is  carried  much 
farther  than  has  even  been  thought  of  here,  or  than  seems  pos- 
sible or  desirable  in  the  future.  To  speak  of  the  arithmetic  of 
these  various  forms  of  schools — for  foresters,  builders,  watch- 
makers, barbers,  and  so  on — would  therefore  be  unprofitable  in 
an  article  like  the  present.  It  will  not,  however,  be  out  of  place 
to  give  an  outline  of  the  work  in  arithmetic  in  the  eighth  school 
year  in  a  girls'  school  in  Munich,  because  this  shows  the  ten- 
dency at  present,  in  one  of  the  most  progressive  cities  of  Europe, 
to  have  arithmetic  touch  the  interests  and  needs  of  the  people. 
The  work  is  as  follows: 

I.     Simple  domestic  bookkeeping. 


The  Work  of  the  Eighth  School  Year  119 

2.  Calculation  of  the  prices  of  foods,  bought  in  large  or  in 
small  quantities,  together  with  the  question  of  discounts. 

3.  Cost  of  meals  for  the  home. 

4.  Daily,  monthly,  and  yearly  supplies  for  the  kitchen,  to- 
gether with  the  keeping  of  kitchen  accounts. 

5.  Simple  measurements  as  needed  in  the  household. 

6.  Food  values  of  different  food  stuffs  as  necessary  for  a 
complete  meal,  with  doubtless  the  application  of  ratio  and  per- 
centage. 

7.  Cost  of  furnishing  a  kitchen. 

8.  Measurements  of  material,  and  the  cost  of  buying,  reno- 
vating, and  washing  clothing  made    of    various    goods,    as    of 
woolen  or  linen. 

9.  Relative  cost  of  different  systems  of  heating. 

10.  Relative  cost  of  different  systems  of  lighting. 

11.  Maintenance  of  the  house,  including  questions  of  rent, 
water,  taxes,  insurance,  and  interest  on  a  mortgage. 

12.  Elementary  commercial  arithmetic,  including  such  general 
topics  as  percentage  and  its  application  in  discount. 

Such  a  course  is  highly  to  be  commended.  It  meets  the  needs 
of  girls  as  we  are  not  meeting  them  in  our  American  schools. 
Indeed  it  becomes  a  serious  question  if,  in  a  subject  like  mathe- 
matics, we  are  not  bound  to  have  separate  classes  for  girls  and 
boys  after  the  seventh  grade  and  through  our  high  school. 

It  may  be  well,  also,  to  consider  the  work  done  in  "  Oberter- 
tia  "  in  a  Prussian  Gymnasium,  corresponding  in  years  to  our 
eighth  grade.  This  correspondence  is  not  exact  in  some  respects, 
because  the  Prussian  school  year  is  somewhat  longer  than  ours, 
but  allowing  three  years  before  entering  the  lowest  class  (Sexta) 
this  becomes  the  eighth  school  year.  There  three  hours  a  week 
are  allowed  to  mathematics,  the  arrangement  allowing  two  to 
geometry  and  one  to  algebra  in  one  week,  and  two  to  algebra 
and  one  to  geometry  the  next  week,  and  so  on.  The  algebra 
includes  equations  of  the  first  degree  with  two  unknown  quan- 
tities, and  the  geometry  finishes  the  treatment  of  the  circle,  finding 
the  value  of  pi.  The  arithmetic  work,  as  we  call  it,  is  practically 
completed  at  the  end  of  the  sixth  school  year  (in  Quarta). 

This  gives  an  idea  of  what  could  be  done  in  America  if  we 
should  care  to  set  about  it.  As  it  is.  the  question  is  a  fair  one 
if  we  would  not  be  justified  in  materially  reducing  the  arith- 


I2O  The  Teaching  of  Arithmetic 

metic  of  Grade  VII  so  as  to  include  a  considerable  part  of  the 
work  of  Grade  VIII,  thus  allowing  an  elementary  course  in 
algebra  in  that  year. 

Algebra.  If  algebra  be  introduced  in  Grade  VIII,  what  is 
the  purpose  and  what  should  be  its  nature?  Aside  from  the 
general  information  thus  given,  and  from  the  discipline  that 
comes  from  this  or  any  other  subject,  there  is  a  need  that  a  few 
years  ago  was  hardly  felt  in  this  country.  Boys  are  apt  to  leave 
school  after  the  work  of  Grade  VIII  is  finished.  They  go  into 
the  shops,  into  trade,  into  various  occupations.  Algebra  was  a 
few  years  ago  of  no  practical  value  to  them,  but  to-day  the 
formula  and  the  graph  of  a  function  are  common  features  in  our 
trade  journals.  Here  then  is  a  suggestion  to  our  schools.  Why 
should  not  elementary  algebra  be  introduced  by  a  study  of  form- 
ulas, so  that  the  simple  algebraic  expressions  of  our  trade  jour- 
nals or  our  artisans'  manuals  can  be  read  easily?  Why  should 
we  not  introduce  graphs  of  functions  very  early,  not  in  compli- 
cated forms  but  as  used  in  the  journals,  the  manuals,  and  the 
workshop  to-day? 

The  author's  views  as  to  details  have  been  set  forth  in  the 
Wentworth-Smith  Vocational  Algebra,  Boston,  1911,  and  in  his 
Algebra  for  Beginners,  Boston,  1906.  The  whole  question  has 
recently  been  discussed  in  a  masterly  little  monograph  written 
by  Mr.  C.  Godfrey,  one  of  the  present  leaders  in  English  edu- 
cation,— The  Algebra  Syllabus  in  the  Secondary  School,  London, 
1911. 

As  to  the  geometry,  the  work  in  mensuration  in  arithmetic 
probably  suffices  for  the  present,  although  it  is  possible  that  we 
may  come  to  adopt  the  German  plan  of  introducing  the  scientific 
treatment  of  the  subject  into  the  elementary  grades  in  the  future. 

Such  are  some  of  the  problems  in  the  teaching  of  arithmetic 
to-day.  Many  are  solved  and  many  still  await  solution  and  are 
occupying  the  attention  of  a  large  number  of  teachers.  It  is 
with  the  hope  of  suggesting  some  of  the  larger  problems  that  this 
book  is  written,  rather  than  with  any  desire  to  treat  the  minor 
details  that  are  sufficiently  discussed  in  any  good  text-book.  If 
the  work  shall  lead  to  sane  experiment,  to  a  conservative  view  of 
the  reforms  to  be  accomplished,  and  to  a  sympathy  with  the 
effort  to  improve  the  problems  of  arithmetic,  it  will  have  served 
its  purpose. 


SOUTHERN    BRANCH 

UNIVERSITY  OF  CALIFORNIA 
LIBRARY 

LOS  ANGELES.  CALIF, 


